Sihangli/scibench-rl huggingface dataset:
Problem:  Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter.
Answer:  672.4  Year
Numeric value:  672.4
Problem: A tank originally contains $100 \mathrm{gal}$ of fresh water. Then water containing $\frac{1}{2} \mathrm{lb}$ of salt per gallon is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture is allowed to leave at the same rate. After $10 \mathrm{~min}$ the process is stopped, and fresh water is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional $10 \mathrm{~min}$.
Answer:  7.42  lb
Numeric value:  7.42
Problem: A young person with no initial capital invests $k$ dollars per year at an annual rate of return $r$. Assume that investments are made continuously and that the return is compounded continuously.
If $r=7.5 \%$, determine $k$ so that $\$ 1$ million will be available for retirement in 40 years.
Answer:  3930  $
Numeric value: 3930
Problem: Consider the initial value problem
$$
y^{\prime \prime}+2 a y^{\prime}+\left(a^2+1\right) y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 .
$$
For $a=1$ find the smallest $T$ such that $|y(t)|<0.1$ for $t>T$.
Answer: 1.8763  
Numeric value: 1.8763
Problem: Consider the initial value problem
$$
y^{\prime \prime}+\gamma y^{\prime}+y=\delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0,
$$
where $\gamma$ is the damping coefficient (or resistance).
Find the time $t_1$ at which the solution attains its maximum value.
Answer:  2.3613  
Numeric value: 2.3613
Problem: Consider the initial value problem
$$
y^{\prime}+\frac{2}{3} y=1-\frac{1}{2} t, \quad y(0)=y_0 .
$$
Find the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis.
Answer:  −1.642876  
Numeric value: −1.642876
Problem: A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportional to the amount currently present. If $Q(t)$ is the amount present at time $t$, then $d Q / d t=-r Q$, where $r>0$ is the decay rate. If $100 \mathrm{mg}$ of thorium-234 decays to $82.04 \mathrm{mg}$ in 1 week, determine the decay rate $r$.
Answer:  0.02828  $\text{day}^{-1}$
Numeric value: 0.02828
Problem: Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200^{\circ} \mathrm{F}$ when freshly poured, and $1 \mathrm{~min}$ later has cooled to $190^{\circ} \mathrm{F}$ in a room at $70^{\circ} \mathrm{F}$, determine when the coffee reaches a temperature of $150^{\circ} \mathrm{F}$.
Answer:  6.07  min
Numeric value:  6.07
Problem: Solve the initial value problem $4 y^{\prime \prime}-y=0, y(0)=2, y^{\prime}(0)=\beta$. Then find $\beta$ so that the solution approaches zero as $t \rightarrow \infty$.
Answer:  -1  
Numeric value:  -1
Problem: Consider the initial value problem (see Example 5)
$$
y^{\prime \prime}+5 y^{\prime}+6 y=0, \quad y(0)=2, \quad y^{\prime}(0)=\beta
$$
where $\beta>0$.
Determine the smallest value of $\beta$ for which $y_m \geq 4$.
Answer:  16.3923  
Numeric value: 16.3923
Problem: A home buyer can afford to spend no more than $\$ 800 /$ month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.
Determine the total interest paid during the term of the mortgage.
Answer:  102,965.21  $
Numeric value: 102,965.21
Problem: Find the fundamental period of the given function:
$$f(x)=\left\{\begin{array}{ll}(-1)^n, & 2 n-1 \leq x<2 n, \\ 1, & 2 n \leq x<2 n+1 ;\end{array} \quad n=0, \pm 1, \pm 2, \ldots\right.$$
Answer:  4  
Numeric value: 4
Problem: A homebuyer wishes to finance the purchase with a \$95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed \$900?
Answer:  9.73  %
Numeric value:  9.73
Problem: A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period. What monthly payment is required if the interest rate is 9%?
Answer: 804.62 $
Numeric value: 804.62
Problem: Let a metallic rod $20 \mathrm{~cm}$ long be heated to a uniform temperature of $100^{\circ} \mathrm{C}$. Suppose that at $t=0$ the ends of the bar are plunged into an ice bath at $0^{\circ} \mathrm{C}$, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Determine the temperature at the center of the bar at time $t=30 \mathrm{~s}$ if the bar is made of silver.
Answer:  35.91  ${ }^{\circ} \mathrm{C}$
Numeric value:  35.91
Problem: Find $\gamma$ so that the solution of the initial value problem $x^2 y^{\prime \prime}-2 y=0, y(1)=1, y^{\prime}(1)=\gamma$ is bounded as $x \rightarrow 0$.
Answer:  2  
Numeric value: 2
Problem: A tank contains 100 gal of water and $50 \mathrm{oz}$ of salt. Water containing a salt concentration of $\frac{1}{4}\left(1+\frac{1}{2} \sin t\right) \mathrm{oz} / \mathrm{gal}$ flows into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture in the tank flows out at the same rate.
The long-time behavior of the solution is an oscillation about a certain constant level. What is the amplitude of the oscillation?
Answer:  0.24995  
Numeric value: 0.24995
Problem: A mass weighing $8 \mathrm{lb}$ stretches a spring 1.5 in. The mass is also attached to a damper with coefficient $\gamma$. Determine the value of $\gamma$ for which the system is critically damped; be sure to give the units for $\gamma$
Answer: 8  $\mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}$
Numeric value: 8
Problem: Your swimming pool containing 60,000 gal of water has been contaminated by $5 \mathrm{~kg}$ of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of $200 \mathrm{gal} / \mathrm{min}$. Find the time $T$ at which the concentration of dye first reaches the value $0.02 \mathrm{~g} / \mathrm{gal}$.
Answer:  7.136  hour
Numeric value: 7.136
Problem: For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity.  If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s.
Answer: $\frac{2}{49}$  
Numeric value: 0.0408
Problem: Consider the initial value problem
$$
3 u^{\prime \prime}-u^{\prime}+2 u=0, \quad u(0)=2, \quad u^{\prime}(0)=0
$$
For $t>0$ find the first time at which $|u(t)|=10$.
Answer:  10.7598  
Numeric value:  10.7598
Problem: Consider the initial value problem
$$
9 y^{\prime \prime}+12 y^{\prime}+4 y=0, \quad y(0)=a>0, \quad y^{\prime}(0)=-1
$$
Find the critical value of $a$ that separates solutions that become negative from those that are always positive.
Answer:  1.5  
Numeric value: 1.5