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Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 1: | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b. |
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: | ## Step 2: First, let's consider the point where x = 2.
We need to equate ax + 3 and x - 5 at x = 2. So, we substitute 2 into both expressions to get a(2) + 3 = 2 - 5. |
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 3: Simplifying the equation a(2) + 3 = 2 - 5, we get 2a + 3 = -3.
Now, we solve for a by subtracting 3 from both sides, resulting in 2a = -6, and then dividing by 2, giving a = -3. |
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 4: Next, let's consider the point where x = -2.
We need to equate x - 5 and 2x - b at x = -2. So, we substitute -2 into both expressions to get -2 - 5 = 2(-2) - b. |
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 5: Simplifying the equation -2 - 5 = 2(-2) - b, we get -7 = -4 - b.
Now, we solve for b by adding 4 to both sides, resulting in -3 = -b, and then multiplying by -1, giving b = 3. |
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 6: Finally, we need to find a + b.
We have already found a = -3 and b = 3, so a + b = -3 + 3. |
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 7: Simplifying a + b = -3 + 3, we get a + b = 0. |
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | ## Step 1: For the function to be continuous, we need to ensure that the different parts of the piecewise function meet at the boundary points.
The boundary points are where x = 2 and x = -2. We need to set the different parts of the function equal to each other at these points and solve for a and b.
## Step 2: First,... | Therefore, the final answer is: $\boxed{0}$. |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$. |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: | ## Step 2: Define the new rectangular formation
When the director increases the number of members in each row by 1 and reduces the number of rows by 2, the new formation has $m + 1$ members in each of $r - 2$ rows, so $n = (m + 1)(r - 2)$. |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 3: Form an equation by equating the two expressions for n
Since the total number of band members $n$ is the same in both formations, we can equate the two expressions and obtain $mr + 2 = (m + 1)(r - 2)$. |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 4: Expand and rearrange the equation
Expanding and rearranging the equation, we get $mr + 2 = mr - 2m + r - 2$. This simplifies to $2m + 2 = r - 2m$ and then to $4m + 2 = r$. |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 5: Express n in terms of m
Substituting the expression $r = 4m + 2$ into the equation $n = mr + 2$, we obtain $n = m(4m + 2) + 2 = 4m^2 + 2m + 2$. |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 6: Determine the constraints on m and n
Since there are less than 100 band members, $n < 100$. We need to find the largest integer value of $n$ that satisfies this condition. |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 7: Find the largest possible integer value of m
The largest possible integer value of $m$ is the largest integer such that $4m^2 + 2m + 2 < 100$. |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 8: Solve the inequality
Solving the inequality $4m^2 + 2m + 2 < 100$, we get $4m^2 + 2m - 98 < 0$. This can be written as $2m^2 + m - 49 < 0$. Using the quadratic formula, we can find the roots of the related equation $2m^2 + m - 49 = 0$ to be $m = \frac{-1 \pm \sqrt{1 + 392}}{4}$. The discriminant is greater t... |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 9: Find the largest possible integer value of m
Since $m$ must be a positive integer, the largest possible value of $m$ is 7. |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 10: Find the corresponding value of n
Substituting $m = 7$ into the expression $n = 4m^2 + 2m + 2$, we obtain $n = 4(7)^2 + 2(7) + 2 = 4(49) + 14 + 2 = 196 + 14 + 2 = 212$. |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | ## Step 1: Define the initial rectangular formation in terms of m and r
Let $n$ be the total number of band members. Initially, the band members are arranged in a rectangular formation with $m$ members in each of $r$ rows, so $n = mr + 2$.
## Step 2: Define the new rectangular formation
When the director increases the... | The final answer is: $\boxed{98}$ |
What is the degree of the polynomial $(4 +5x^3 +100 +2\pi x^4 + \sqrt{10}x^4 +9)$? | 4 | ## Step 1: Identify the term with the highest power of x in the polynomial.
The polynomial contains several terms with different powers of x: $x^3$, $x^4$, and constant terms.
## Step 2: Determine the degree of the polynomial based on the highest power of x.
The term with the highest power of x is $2\pi x^4$, which ha... | ## Step 1: | ## Step 1: Identify the term with the highest power of x in the polynomial.
The polynomial contains several terms with different powers of x: $x^3$, $x^4$, and constant terms. |
What is the degree of the polynomial $(4 +5x^3 +100 +2\pi x^4 + \sqrt{10}x^4 +9)$? | 4 | ## Step 1: Identify the term with the highest power of x in the polynomial.
The polynomial contains several terms with different powers of x: $x^3$, $x^4$, and constant terms.
## Step 2: Determine the degree of the polynomial based on the highest power of x.
The term with the highest power of x is $2\pi x^4$, which ha... | ## Step 1: Identify the term with the highest power of x in the polynomial.
The polynomial contains several terms with different powers of x: $x^3$, $x^4$, and constant terms.
## Step 2: | ## Step 2: Determine the degree of the polynomial based on the highest power of x.
The term with the highest power of x is $2\pi x^4$, which has a power of 4. Therefore, the degree of the polynomial is 4. |
What is the degree of the polynomial $(4 +5x^3 +100 +2\pi x^4 + \sqrt{10}x^4 +9)$? | 4 | ## Step 1: Identify the term with the highest power of x in the polynomial.
The polynomial contains several terms with different powers of x: $x^3$, $x^4$, and constant terms.
## Step 2: Determine the degree of the polynomial based on the highest power of x.
The term with the highest power of x is $2\pi x^4$, which ha... | ## Step 1: Identify the term with the highest power of x in the polynomial.
The polynomial contains several terms with different powers of x: $x^3$, $x^4$, and constant terms.
## Step 2: Determine the degree of the polynomial based on the highest power of x.
The term with the highest power of x is $2\pi x^4$, which ha... | Therefore, the final answer is: $\boxed{4}$. |
Evaluate $\left\lceil3\left(6-\frac12\right)\right\rceil$. | 17 | ## Step 1: Evaluate the expression inside the parenthesis
First, we need to calculate $6-\frac12$. This is equal to $6-0.5$, which equals $5.5$.
## Step 2: Multiply the result by 3
Next, we multiply the result from step 1 by 3. So, $3\times 5.5 = 16.5$.
## Step 3: Apply the ceiling function
The ceiling function round... | ## Step 1: | ## Step 1: Evaluate the expression inside the parenthesis
First, we need to calculate $6-\frac12$. This is equal to $6-0.5$, which equals $5.5$. |
Evaluate $\left\lceil3\left(6-\frac12\right)\right\rceil$. | 17 | ## Step 1: Evaluate the expression inside the parenthesis
First, we need to calculate $6-\frac12$. This is equal to $6-0.5$, which equals $5.5$.
## Step 2: Multiply the result by 3
Next, we multiply the result from step 1 by 3. So, $3\times 5.5 = 16.5$.
## Step 3: Apply the ceiling function
The ceiling function round... | ## Step 1: Evaluate the expression inside the parenthesis
First, we need to calculate $6-\frac12$. This is equal to $6-0.5$, which equals $5.5$.
## Step 2: | ## Step 2: Multiply the result by 3
Next, we multiply the result from step 1 by 3. So, $3\times 5.5 = 16.5$. |
Evaluate $\left\lceil3\left(6-\frac12\right)\right\rceil$. | 17 | ## Step 1: Evaluate the expression inside the parenthesis
First, we need to calculate $6-\frac12$. This is equal to $6-0.5$, which equals $5.5$.
## Step 2: Multiply the result by 3
Next, we multiply the result from step 1 by 3. So, $3\times 5.5 = 16.5$.
## Step 3: Apply the ceiling function
The ceiling function round... | ## Step 1: Evaluate the expression inside the parenthesis
First, we need to calculate $6-\frac12$. This is equal to $6-0.5$, which equals $5.5$.
## Step 2: Multiply the result by 3
Next, we multiply the result from step 1 by 3. So, $3\times 5.5 = 16.5$.
## Step 3: | ## Step 3: Apply the ceiling function
The ceiling function rounds up to the nearest whole number. Therefore, $\lceil 16.5 \rceil = 17$. |
Evaluate $\left\lceil3\left(6-\frac12\right)\right\rceil$. | 17 | ## Step 1: Evaluate the expression inside the parenthesis
First, we need to calculate $6-\frac12$. This is equal to $6-0.5$, which equals $5.5$.
## Step 2: Multiply the result by 3
Next, we multiply the result from step 1 by 3. So, $3\times 5.5 = 16.5$.
## Step 3: Apply the ceiling function
The ceiling function round... | ## Step 1: Evaluate the expression inside the parenthesis
First, we need to calculate $6-\frac12$. This is equal to $6-0.5$, which equals $5.5$.
## Step 2: Multiply the result by 3
Next, we multiply the result from step 1 by 3. So, $3\times 5.5 = 16.5$.
## Step 3: Apply the ceiling function
The ceiling function round... | Therefore, the final answer is: $\boxed{17}$. |
Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | 6 | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 1: | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work. |
Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | 6 | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: | ## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not work, $30 is subtracted from his earnings. Therefore, his total earnings can be represented by 60x - 30y = 660. |
Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | 6 | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 3: Express y in terms of x
From the equation x + y = 20, we can express y as y = 20 - x. |
Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | 6 | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 4: Substitute y in the earnings equation
Substitute y = 20 - x into the earnings equation 60x - 30y = 660 to get 60x - 30(20 - x) = 660. |
Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | 6 | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 5: Solve for x
Expanding the equation gives 60x - 600 + 30x = 660. Combining like terms yields 90x - 600 = 660. Adding 600 to both sides gives 90x = 1260. Dividing by 90 gives x = 14. |
Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | 6 | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 6: Find the value of y
Since y = 20 - x and x = 14, we have y = 20 - 14 = 6. |
Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | 6 | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | ## Step 1: Define the variables
Let's define two variables: x as the number of days Sam worked and y as the number of days he did not work.
## Step 2: Set up the equations
Since the total number of days is 20, we have x + y = 20. We also know that for every day Sam works, he earns $60, and for every day he does not wo... | Therefore, the final answer is: $\boxed{6}$ |
Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. | (3, -1) | ## Step 1: Complete the square for the x terms
The equation can be rewritten as $(x^2 - 6x) + y^2 + 2y = 9$. Completing the square for the x terms, we get $(x^2 - 6x + 9) - 9 + y^2 + 2y = 9$, which simplifies to $(x-3)^2 + y^2 + 2y = 18$.
## Step 2: Complete the square for the y terms
Now, completing the square for th... | ## Step 1: | ## Step 1: Complete the square for the x terms
The equation can be rewritten as $(x^2 - 6x) + y^2 + 2y = 9$. Completing the square for the x terms, we get $(x^2 - 6x + 9) - 9 + y^2 + 2y = 9$, which simplifies to $(x-3)^2 + y^2 + 2y = 18$. |
Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. | (3, -1) | ## Step 1: Complete the square for the x terms
The equation can be rewritten as $(x^2 - 6x) + y^2 + 2y = 9$. Completing the square for the x terms, we get $(x^2 - 6x + 9) - 9 + y^2 + 2y = 9$, which simplifies to $(x-3)^2 + y^2 + 2y = 18$.
## Step 2: Complete the square for the y terms
Now, completing the square for th... | ## Step 1: Complete the square for the x terms
The equation can be rewritten as $(x^2 - 6x) + y^2 + 2y = 9$. Completing the square for the x terms, we get $(x^2 - 6x + 9) - 9 + y^2 + 2y = 9$, which simplifies to $(x-3)^2 + y^2 + 2y = 18$.
## Step 2: | ## Step 2: Complete the square for the y terms
Now, completing the square for the y terms, we get $(x-3)^2 + (y^2 + 2y + 1) - 1 = 18$. This simplifies to $(x-3)^2 + (y+1)^2 = 19$. |
Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. | (3, -1) | ## Step 1: Complete the square for the x terms
The equation can be rewritten as $(x^2 - 6x) + y^2 + 2y = 9$. Completing the square for the x terms, we get $(x^2 - 6x + 9) - 9 + y^2 + 2y = 9$, which simplifies to $(x-3)^2 + y^2 + 2y = 18$.
## Step 2: Complete the square for the y terms
Now, completing the square for th... | ## Step 1: Complete the square for the x terms
The equation can be rewritten as $(x^2 - 6x) + y^2 + 2y = 9$. Completing the square for the x terms, we get $(x^2 - 6x + 9) - 9 + y^2 + 2y = 9$, which simplifies to $(x-3)^2 + y^2 + 2y = 18$.
## Step 2: Complete the square for the y terms
Now, completing the square for th... | ## Step 3: Identify the center from the standard form of the circle equation
The equation $(x-3)^2 + (y+1)^2 = 19$ is in the standard form of a circle equation, $(x-h)^2 + (y-k)^2 = r^2$. Comparing the two, we can see that $h=3$ and $k=-1$. |
Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. | (3, -1) | ## Step 1: Complete the square for the x terms
The equation can be rewritten as $(x^2 - 6x) + y^2 + 2y = 9$. Completing the square for the x terms, we get $(x^2 - 6x + 9) - 9 + y^2 + 2y = 9$, which simplifies to $(x-3)^2 + y^2 + 2y = 18$.
## Step 2: Complete the square for the y terms
Now, completing the square for th... | ## Step 1: Complete the square for the x terms
The equation can be rewritten as $(x^2 - 6x) + y^2 + 2y = 9$. Completing the square for the x terms, we get $(x^2 - 6x + 9) - 9 + y^2 + 2y = 9$, which simplifies to $(x-3)^2 + y^2 + 2y = 18$.
## Step 2: Complete the square for the y terms
Now, completing the square for th... | Therefore, the final answer is: $\boxed{(3, -1)}$. |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 1: | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$. |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: | ## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$. |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: | ## Step 3: Replace the powers in the expression
We now have $\frac{16+2(25)}{6}$. |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 4: Multiply 2 and 25
$2(25) = 50$. |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 5: Replace the product in the expression
The expression now becomes $\frac{16+50}{6}$. |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 6: Add the numbers in the numerator
$16+50 = 66$. |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 7: Replace the sum in the expression
The expression now is $\frac{66}{6}$. |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 8: Divide the numerator by the denominator
$\frac{66}{6} = 11$. |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | ## Step 1: Substitute the given values of x and y into the expression
We have x = 2 and y = 5, so we will replace these values in the expression $\frac{x^4+2y^2}{6}$ to get $\frac{(2)^4+2(5)^2}{6}$.
## Step 2: Evaluate the powers of x and y
Calculate the powers of x and y: $2^4 = 16$ and $5^2 = 25$.
## Step 3: Replac... | Therefore, the final answer is: $\boxed{11}$. |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 2: Find the correct difference for the column of squares.
The column of squares forms an arithmetic sequence, and the difference between 18 and -17 is 35. However, looking back, the difference between 14 and 18 is 4, which suggests that there is a common difference in this column of 4, not -9. To check, let's l... |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 3: Find the value of N using the column of squares.
Since the difference between 18 and -17 is 35, the sequence in the right column would be -17, -13, -9, -5, -1, 3, 7, 11, 15, 19, 23, 27, 31, 35. Since the sequence starts at -17 and the difference between terms is 4, we can find the term in the top row of the ... |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 4: Find the common difference in the row of squares.
However, looking at the row of squares, the difference from 21 to 14 is 7. We can continue this pattern to find the next number in the row. We subtract 7 from 14 to get 7, subtract 7 from 7 to get -0 (or 0), subtract 7 from 0 to get -7, and subtract 7 from -7... |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 5: Find the term in the row that corresponds to the term -17 in the column.
Since we are looking for the term in the row that corresponds to the term -17, we need to go down 4 from 21 to get the term in the row that corresponds to -17. 21 - 4 = 17. |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 6: Find the value of N.
However, we are not looking for 17, we are looking for N, which is the term in the row that corresponds to the term -17 in the column. Since we know the term in the row that corresponds to -17 is -7 (21 - 7 = 14 and 14 - 7 = 7, 7 - 7 = 0, 0 - 7 = -7, and -7 - 7 = -14, so -17 is two terms... |
The sequence of integers in the row of squares and in each of the two columns of squares form three distinct arithmetic sequences. What is the value of $N$?
[asy]
unitsize(0.35inch);
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,... | -7 | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... | ## Step 1: Analyze the arithmetic sequences in the column of squares.
Looking at the column on the left of the figure, we can see that the common difference between consecutive terms is $-9$ (from 21 to 14 is a decrease of 7, and from 14 to 18 is an increase of 4, but there is another set of terms to consider), so we'l... |
The final answer is: $\boxed{-7}$ |
The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? | 0 | ## Step 1: Recall the formula for the center of a circle given its diameter
The center of a circle is the midpoint of its diameter. The midpoint formula is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of the diameter.
## Step 2: Apply the midpoint formula to the given... | ## Step 1: | ## Step 1: Recall the formula for the center of a circle given its diameter
The center of a circle is the midpoint of its diameter. The midpoint formula is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of the diameter. |
The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? | 0 | ## Step 1: Recall the formula for the center of a circle given its diameter
The center of a circle is the midpoint of its diameter. The midpoint formula is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of the diameter.
## Step 2: Apply the midpoint formula to the given... | ## Step 1: Recall the formula for the center of a circle given its diameter
The center of a circle is the midpoint of its diameter. The midpoint formula is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of the diameter.
## Step 2: | ## Step 2: Apply the midpoint formula to the given endpoints
Using the midpoint formula with $(x_1, y_1) = (9, -5)$ and $(x_2, y_2) = (-3, -1)$, we get: $(\frac{9 + (-3)}{2}, \frac{-5 + (-1)}{2}) = (\frac{6}{2}, \frac{-6}{2}) = (3, -3)$. |
The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? | 0 | ## Step 1: Recall the formula for the center of a circle given its diameter
The center of a circle is the midpoint of its diameter. The midpoint formula is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of the diameter.
## Step 2: Apply the midpoint formula to the given... | ## Step 1: Recall the formula for the center of a circle given its diameter
The center of a circle is the midpoint of its diameter. The midpoint formula is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of the diameter.
## Step 2: Apply the midpoint formula to the given... | ## Step 3: Find the sum of the coordinates of the center of the circle
To find the sum of the coordinates of the center, we simply add them together: $3 + (-3) = 0$. |
The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? | 0 | ## Step 1: Recall the formula for the center of a circle given its diameter
The center of a circle is the midpoint of its diameter. The midpoint formula is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of the diameter.
## Step 2: Apply the midpoint formula to the given... | ## Step 1: Recall the formula for the center of a circle given its diameter
The center of a circle is the midpoint of its diameter. The midpoint formula is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of the diameter.
## Step 2: Apply the midpoint formula to the given... | Therefore, the final answer is: $\boxed{0}$. |
Berengere and her American foreign-exchange student Emily are at a bakery in Paris that accepts both euros and American dollars. They want to buy a cake, but neither of them has enough money. If the cake costs 6 euros and Emily has an American five-dollar bill, how many euros does Berengere need to contribute to the co... | 2 \text{ euros} | ## Step 1: Convert the value of Emily's money to euros
Since 1 euro is equal to 1.25 USD, Emily's $5 is equivalent to 5 / 1.25 = 4 euros.
## Step 2: Calculate the amount Berengere needs to contribute
The cake costs 6 euros, and Emily has 4 euros. Berengere needs to contribute the remaining amount, which is 6 - 4 = 2 e... | ## Step 1: | ## Step 1: Convert the value of Emily's money to euros
Since 1 euro is equal to 1.25 USD, Emily's $5 is equivalent to 5 / 1.25 = 4 euros. |
Berengere and her American foreign-exchange student Emily are at a bakery in Paris that accepts both euros and American dollars. They want to buy a cake, but neither of them has enough money. If the cake costs 6 euros and Emily has an American five-dollar bill, how many euros does Berengere need to contribute to the co... | 2 \text{ euros} | ## Step 1: Convert the value of Emily's money to euros
Since 1 euro is equal to 1.25 USD, Emily's $5 is equivalent to 5 / 1.25 = 4 euros.
## Step 2: Calculate the amount Berengere needs to contribute
The cake costs 6 euros, and Emily has 4 euros. Berengere needs to contribute the remaining amount, which is 6 - 4 = 2 e... | ## Step 1: Convert the value of Emily's money to euros
Since 1 euro is equal to 1.25 USD, Emily's $5 is equivalent to 5 / 1.25 = 4 euros.
## Step 2: | ## Step 2: Calculate the amount Berengere needs to contribute
The cake costs 6 euros, and Emily has 4 euros. Berengere needs to contribute the remaining amount, which is 6 - 4 = 2 euros. |
Berengere and her American foreign-exchange student Emily are at a bakery in Paris that accepts both euros and American dollars. They want to buy a cake, but neither of them has enough money. If the cake costs 6 euros and Emily has an American five-dollar bill, how many euros does Berengere need to contribute to the co... | 2 \text{ euros} | ## Step 1: Convert the value of Emily's money to euros
Since 1 euro is equal to 1.25 USD, Emily's $5 is equivalent to 5 / 1.25 = 4 euros.
## Step 2: Calculate the amount Berengere needs to contribute
The cake costs 6 euros, and Emily has 4 euros. Berengere needs to contribute the remaining amount, which is 6 - 4 = 2 e... | ## Step 1: Convert the value of Emily's money to euros
Since 1 euro is equal to 1.25 USD, Emily's $5 is equivalent to 5 / 1.25 = 4 euros.
## Step 2: Calculate the amount Berengere needs to contribute
The cake costs 6 euros, and Emily has 4 euros. Berengere needs to contribute the remaining amount, which is 6 - 4 = 2 e... | Therefore, the final answer is: $\boxed{2}$. |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 1: | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$. |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: | ## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$ |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: | ## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$ |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: | ## Step 4: Evaluate the cube roots
$\sqrt[3]{9} = 2.0801...$ and $\sqrt[3]{3} = 1.4422...$ |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 5: Multiply the results
$2.0801... \cdot 1.4422... = 3$ |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | However, since we are dealing with cube roots, let's try to simplify the problem using algebraic manipulations instead of decimal approximations. |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 6: Rewrite the expression with algebraic manipulations
Let $x = \sqrt[3]{1+8} = \sqrt[3]{9}$ and $y = \sqrt[3]{1+\sqrt[3]{8}} = \sqrt[3]{3}$.
So, the expression becomes $x \cdot y$. |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 7: Evaluate the expression $x \cdot y$ algebraically
$x \cdot y = \sqrt[3]{9} \cdot \sqrt[3]{3} = \sqrt[3]{9 \cdot 3} = \sqrt[3]{27}$ |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 8: Evaluate the cube root of 27
$\sqrt[3]{27} = 3$ because $3^3 = 27$ |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | ## Step 1: Evaluate the innermost cube root
$\sqrt[3]{8} = 2$ because $2^3 = 8$.
## Step 2: Substitute the innermost cube root into the expression
$\sqrt[3]{1+8} \cdot \sqrt[3]{1+2}$
## Step 3: Simplify the expressions inside the cube roots
$\sqrt[3]{9} \cdot \sqrt[3]{3}$
## Step 4: Evaluate the cube roots
$\sqrt[3]... | Therefore, the final answer is: $\boxed{3}$. |
Let $f(x)=x^3+3$ and $g(x) = 2x^2 + 2x +1$. What is $g(f(-2))$? | 41 | ## Step 1: Evaluate the inner function f(-2) by substituting x with -2 in f(x) = x^3 + 3.
f(-2) = (-2)^3 + 3 = -8 + 3 = -5
## Step 2: Now, substitute the value of f(-2) into g(x) = 2x^2 + 2x + 1 to find g(f(-2)).
g(f(-2)) = g(-5) = 2(-5)^2 + 2(-5) + 1 = 2(25) - 10 + 1 = 50 - 10 + 1 = 41
Therefore, the final answer is... | ## Step 1: | ## Step 1: Evaluate the inner function f(-2) by substituting x with -2 in f(x) = x^3 + 3.
f(-2) = (-2)^3 + 3 = -8 + 3 = -5 |
Let $f(x)=x^3+3$ and $g(x) = 2x^2 + 2x +1$. What is $g(f(-2))$? | 41 | ## Step 1: Evaluate the inner function f(-2) by substituting x with -2 in f(x) = x^3 + 3.
f(-2) = (-2)^3 + 3 = -8 + 3 = -5
## Step 2: Now, substitute the value of f(-2) into g(x) = 2x^2 + 2x + 1 to find g(f(-2)).
g(f(-2)) = g(-5) = 2(-5)^2 + 2(-5) + 1 = 2(25) - 10 + 1 = 50 - 10 + 1 = 41
Therefore, the final answer is... | ## Step 1: Evaluate the inner function f(-2) by substituting x with -2 in f(x) = x^3 + 3.
f(-2) = (-2)^3 + 3 = -8 + 3 = -5
## Step 2: | ## Step 2: Now, substitute the value of f(-2) into g(x) = 2x^2 + 2x + 1 to find g(f(-2)).
g(f(-2)) = g(-5) = 2(-5)^2 + 2(-5) + 1 = 2(25) - 10 + 1 = 50 - 10 + 1 = 41 |
Let $f(x)=x^3+3$ and $g(x) = 2x^2 + 2x +1$. What is $g(f(-2))$? | 41 | ## Step 1: Evaluate the inner function f(-2) by substituting x with -2 in f(x) = x^3 + 3.
f(-2) = (-2)^3 + 3 = -8 + 3 = -5
## Step 2: Now, substitute the value of f(-2) into g(x) = 2x^2 + 2x + 1 to find g(f(-2)).
g(f(-2)) = g(-5) = 2(-5)^2 + 2(-5) + 1 = 2(25) - 10 + 1 = 50 - 10 + 1 = 41
Therefore, the final answer is... | ## Step 1: Evaluate the inner function f(-2) by substituting x with -2 in f(x) = x^3 + 3.
f(-2) = (-2)^3 + 3 = -8 + 3 = -5
## Step 2: Now, substitute the value of f(-2) into g(x) = 2x^2 + 2x + 1 to find g(f(-2)).
g(f(-2)) = g(-5) = 2(-5)^2 + 2(-5) + 1 = 2(25) - 10 + 1 = 50 - 10 + 1 = 41
| Therefore, the final answer is: $\boxed{41}$. |
Let \[f(x) =
\begin{cases}
x/2 &\quad \text{if } x \text{ is even}, \\
3x+1 &\quad \text{if } x \text{ is odd}.
\end{cases}
\]What is $f(f(f(f(1))))$? | 4 | ## Step 1: Evaluate the innermost function f(1).
Since 1 is odd, we use the second case of the function: f(1) = 3(1) + 1 = 4.
## Step 2: Evaluate the next function f(f(1)).
Now that we know f(1) = 4, and 4 is even, we use the first case of the function: f(f(1)) = f(4) = 4/2 = 2.
## Step 3: Evaluate the next function ... | ## Step 1: | ## Step 1: Evaluate the innermost function f(1).
Since 1 is odd, we use the second case of the function: f(1) = 3(1) + 1 = 4. |
Let \[f(x) =
\begin{cases}
x/2 &\quad \text{if } x \text{ is even}, \\
3x+1 &\quad \text{if } x \text{ is odd}.
\end{cases}
\]What is $f(f(f(f(1))))$? | 4 | ## Step 1: Evaluate the innermost function f(1).
Since 1 is odd, we use the second case of the function: f(1) = 3(1) + 1 = 4.
## Step 2: Evaluate the next function f(f(1)).
Now that we know f(1) = 4, and 4 is even, we use the first case of the function: f(f(1)) = f(4) = 4/2 = 2.
## Step 3: Evaluate the next function ... | ## Step 1: Evaluate the innermost function f(1).
Since 1 is odd, we use the second case of the function: f(1) = 3(1) + 1 = 4.
## Step 2: | ## Step 2: Evaluate the next function f(f(1)).
Now that we know f(1) = 4, and 4 is even, we use the first case of the function: f(f(1)) = f(4) = 4/2 = 2. |
Let \[f(x) =
\begin{cases}
x/2 &\quad \text{if } x \text{ is even}, \\
3x+1 &\quad \text{if } x \text{ is odd}.
\end{cases}
\]What is $f(f(f(f(1))))$? | 4 | ## Step 1: Evaluate the innermost function f(1).
Since 1 is odd, we use the second case of the function: f(1) = 3(1) + 1 = 4.
## Step 2: Evaluate the next function f(f(1)).
Now that we know f(1) = 4, and 4 is even, we use the first case of the function: f(f(1)) = f(4) = 4/2 = 2.
## Step 3: Evaluate the next function ... | ## Step 1: Evaluate the innermost function f(1).
Since 1 is odd, we use the second case of the function: f(1) = 3(1) + 1 = 4.
## Step 2: Evaluate the next function f(f(1)).
Now that we know f(1) = 4, and 4 is even, we use the first case of the function: f(f(1)) = f(4) = 4/2 = 2.
## Step 3: | ## Step 3: Evaluate the next function f(f(f(1))).
Since f(f(1)) = 2, and 2 is even, we again use the first case of the function: f(f(f(1))) = f(2) = 2/2 = 1. |
Let \[f(x) =
\begin{cases}
x/2 &\quad \text{if } x \text{ is even}, \\
3x+1 &\quad \text{if } x \text{ is odd}.
\end{cases}
\]What is $f(f(f(f(1))))$? | 4 | ## Step 1: Evaluate the innermost function f(1).
Since 1 is odd, we use the second case of the function: f(1) = 3(1) + 1 = 4.
## Step 2: Evaluate the next function f(f(1)).
Now that we know f(1) = 4, and 4 is even, we use the first case of the function: f(f(1)) = f(4) = 4/2 = 2.
## Step 3: Evaluate the next function ... | ## Step 1: Evaluate the innermost function f(1).
Since 1 is odd, we use the second case of the function: f(1) = 3(1) + 1 = 4.
## Step 2: Evaluate the next function f(f(1)).
Now that we know f(1) = 4, and 4 is even, we use the first case of the function: f(f(1)) = f(4) = 4/2 = 2.
## Step 3: Evaluate the next function ... | ## Step 4: Evaluate the final function f(f(f(f(1))).
Since f(f(f(1))) = 1, and 1 is odd, we once again use the second case of the function: f(f(f(f(1)))) = f(1) = 3(1) + 1 = 4. |
Let \[f(x) =
\begin{cases}
x/2 &\quad \text{if } x \text{ is even}, \\
3x+1 &\quad \text{if } x \text{ is odd}.
\end{cases}
\]What is $f(f(f(f(1))))$? | 4 | ## Step 1: Evaluate the innermost function f(1).
Since 1 is odd, we use the second case of the function: f(1) = 3(1) + 1 = 4.
## Step 2: Evaluate the next function f(f(1)).
Now that we know f(1) = 4, and 4 is even, we use the first case of the function: f(f(1)) = f(4) = 4/2 = 2.
## Step 3: Evaluate the next function ... | ## Step 1: Evaluate the innermost function f(1).
Since 1 is odd, we use the second case of the function: f(1) = 3(1) + 1 = 4.
## Step 2: Evaluate the next function f(f(1)).
Now that we know f(1) = 4, and 4 is even, we use the first case of the function: f(f(1)) = f(4) = 4/2 = 2.
## Step 3: Evaluate the next function ... | Therefore, the final answer is: $\boxed{4}$. |
The greatest integer function, $\lfloor x\rfloor$, denotes the largest integer less than or equal to $x$. For example, $\lfloor3.5\rfloor=3$, $\lfloor\pi\rfloor=3$ and $\lfloor -\pi\rfloor=-4$. Find the sum of the three smallest positive solutions to $x-\lfloor x\rfloor=\frac1{\lfloor x\rfloor}.$ Express your answer as... | 10\frac{1}{12} | ## Step 1: Analyze the equation $x-\lfloor x\rfloor=\frac{1}{\lfloor x\rfloor}$.
The equation involves the greatest integer function, indicating that the fractional part of $x$ should be equal to $\frac{1}{\lfloor x\rfloor}$.
## Step 2: Consider possible values of $\lfloor x\rfloor$.
We can start with $\lfloor x\rfloo... | ## Step 1: | ## Step 1: Analyze the equation $x-\lfloor x\rfloor=\frac{1}{\lfloor x\rfloor}$.
The equation involves the greatest integer function, indicating that the fractional part of $x$ should be equal to $\frac{1}{\lfloor x\rfloor}$. |
The greatest integer function, $\lfloor x\rfloor$, denotes the largest integer less than or equal to $x$. For example, $\lfloor3.5\rfloor=3$, $\lfloor\pi\rfloor=3$ and $\lfloor -\pi\rfloor=-4$. Find the sum of the three smallest positive solutions to $x-\lfloor x\rfloor=\frac1{\lfloor x\rfloor}.$ Express your answer as... | 10\frac{1}{12} | ## Step 1: Analyze the equation $x-\lfloor x\rfloor=\frac{1}{\lfloor x\rfloor}$.
The equation involves the greatest integer function, indicating that the fractional part of $x$ should be equal to $\frac{1}{\lfloor x\rfloor}$.
## Step 2: Consider possible values of $\lfloor x\rfloor$.
We can start with $\lfloor x\rfloo... | ## Step 1: Analyze the equation $x-\lfloor x\rfloor=\frac{1}{\lfloor x\rfloor}$.
The equation involves the greatest integer function, indicating that the fractional part of $x$ should be equal to $\frac{1}{\lfloor x\rfloor}$.
## Step 2: | ## Step 2: Consider possible values of $\lfloor x\rfloor$.
We can start with $\lfloor x\rfloor=1$. However, this is impossible since $\frac{1}{1}=1$ but $0\leq x-\lfloor x\rfloor<1$. So, we try $\lfloor x\rfloor=2$. We see that $\frac{1}{2}\leq x-\lfloor x\rfloor<1$, which is a possibility. |
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