ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Preamble: The following subproblems refer to the differential equation $\ddot{x}+b \dot{x}+x=0$.\\

Subproblem 0: What is the characteristic polynomial $p(s)$ of $\ddot{x}+b \dot{x}+x=0$?


Solution: The characteristic polynomial is $p(s)=\boxed{s^{2}+b s+1}$.

Final answer: The final answer is s^{2}+b s+1. I hope it is correct.

Subproblem 1: For what value of $b$ does $\ddot{x}+b \dot{x}+x=0$ exhibit critical damping?
Answer: 2
Problem: Find the general (complex-valued) solution of the differential equation $\dot{z}+2 z=e^{2 i t}$, using $C$ to stand for any complex-valued integration constants which may arise.
Answer: \frac{e^{2it}}{(2+2i)}+Ce^{-2t}
Problem: Preamble: Consider the first-order system
\[
\tau \dot{y}+y=u
\]
driven with a unit step from zero initial conditions. The input to this system is \(u\) and the output is \(y\). 

Derive and expression for the settling time \(t_{s}\), where the settling is to within an error \(\pm \Delta\) from the final value of 1.
Answer: -\tau\ln\Delta
Problem: Preamble: Consider the first-order system
\[
\tau \dot{y}+y=u
\]
driven with a unit step from zero initial conditions. The input to this system is \(u\) and the output is \(y\). 

Subproblem 0: Derive and expression for the settling time \(t_{s}\), where the settling is to within an error \(\pm \Delta\) from the final value of 1.


Solution: Rise and Settling Times.  We are given the first-order transfer function
\[
H(s)=\frac{1}{\tau s+1}
\]
The response to a unit step with zero initial conditions will be \(y(t)=1-e^{-t / \tau}\). To determine the amount of time it take \(y\) to settle to within \(\Delta\) of its final value, we want to find the time \(t_{s}\) such that \(y\left(t_{s}\right)=1-\Delta\). Thus, we obtain
\[
\begin{aligned}
&\Delta=e^{-t_{s} / \tau} \\
&t_{s}=\boxed{-\tau \ln \Delta}
\end{aligned}
\]

Final answer: The final answer is -\tau \ln \Delta. I hope it is correct.

Subproblem 1: Derive an expression for the \(10-90 \%\) rise time \(t_{r}\) in terms of $\tau$.
Answer: 2.2\tau
Problem: Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

$y(t)=e^{-a t}$
Answer: \frac{1}{s+a}
Problem: Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :

Subproblem 0: \[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]


Solution: We can simplify with partial fractions:
\[
Y(s)=\frac{1}{(s+a)(s+b)}=\frac{C}{s+a}+\frac{D}{s+b}
\]
find the constants \(C\) and \(D\) by setting \(s=-a\) and \(s=-b\)
\[
\begin{aligned}
\frac{1}{(s+a)(s+b)} &=\frac{C}{s+a}+\frac{D}{s+b} \\
1 &=C(s+b)+D(s+a) \\
C &=\frac{1}{b-a} \\
D &=\frac{1}{a-b}
\end{aligned}
\]
therefore
\[
Y(s)=\frac{1}{b-a} \frac{1}{s+a}-\frac{1}{b-a} \frac{1}{s+b}
\]
By looking up the inverse Laplace Transform of \(\frac{1}{s+b}\), we find the total solution \(y(t)\)
\[
y(t)=\boxed{\frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right)}
\]

Final answer: The final answer is \frac{1}{b-a}\left(e^{-a t}-e^{-b t}\right). I hope it is correct.

Subproblem 1: \[
Y(s)=\frac{s}{\frac{s^{2}}{\omega_{n}^{2}}+\frac{2 \zeta}{\omega_{n}} s+1}
\]
You may assume that $\zeta < 1$.
Answer: \omega_{n}^{2}e^{-\zeta\omega_{n}t}\cos(\omega_{n}\sqrt{1-\zeta^{2}}t)-\frac{\zeta\omega_{n}^{2}}{\sqrt{1-\zeta^{2}}}e^{-\zeta\omega_{n}t}\sin(\omega_{n}\sqrt{1-\zeta^{2}}t)
Problem: A signal \(x(t)\) is given by
\[
x(t)=\left(e^{-t}-e^{-1}\right)\left(u_{s}(t)-u_{s}(t-1)\right)
\]
Calculate its Laplace transform \(X(s)\). Make sure to clearly show the steps in your calculation.
Answer: \frac{1}{s+1}-\frac{e^{-1}}{s}-\frac{e^{-1}e^{-s}}{s+1}+\frac{e^{-1}e^{-s}}{s}
Problem: Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]

Subproblem 0: What is the time constant for this system?


Solution: We find the homogenous solution, solving:
\[
\dot{y}+5 y=0
\]
by trying a solution of the form $y=A \cdot e^{s, t}$.
Calculation:
\[
\dot{y}=A \cdot s \cdot e^{s \cdot t} \mid \Rightarrow A \cdot s \cdot e^{s t}+5 A \cdot e^{s t}=0
\]
yields that $s=-5$, meaning the solution is $y=A \cdot e^{-5 \cdot t}=A \cdot e^{-t / \tau}$, meaning $\tau = \boxed{0.2}$.

Final answer: The final answer is 0.2. I hope it is correct.

Subproblem 1: If \(u=10\), what is the final or steady-state value for \(y(t)\)?
Answer: 20
Problem: A signal \(w(t)\) is defined as
\[
w(t)=u_{s}(t)-u_{s}(t-T)
\]
where \(T\) is a fixed time in seconds and \(u_{s}(t)\) is the unit step. Compute the Laplace transform \(W(s)\) of \(w(t)\). Show your work.
Answer: \frac{1}{s}-\frac{1}{s}e^{-sT}
Problem: Preamble: Assume that we apply a unit step in force separately to a mass \(m\), a dashpot \(c\), and a spring \(k\). The mass moves in inertial space. The spring and dashpot have one end connected to inertial space (reference velocity \(=0\) ), and the force is applied to the other end.  Assume zero initial velocity and position for the elements.
Recall that the unit step function \(u_{S}(t)\) is defined as \(u_{S}(t)=0 ; t<0\) and \(u_{S}(t)=1 ; t \geq 0\). We will also find it useful to introduce the unit impulse function \(\delta(t)\) which can be defined via
\[
u_{S}(t)=\int_{-\infty}^{t} \delta(\tau) d \tau
\]
This means that we can also view the unit impulse as the derivative of the unit step:
\[
\delta(t)=\frac{d u_{S}(t)}{d t}
\]

Solve for the resulting velocity of the mass.
Answer: \frac{1}{m}t
Problem: Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

Subproblem 0: $y(t)=e^{-a t}$


Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]

Final answer: The final answer is \frac{1}{s+a}. I hope it is correct.

Subproblem 1: $y(t)=e^{-\sigma t} \sin \omega_{d} t$


Solution: \[
Y(s)=\boxed{\frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}}
\]

Final answer: The final answer is \frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}. I hope it is correct.

Subproblem 2: $y(t)=e^{-\sigma t} \cos \omega_{d} t$
Answer: \frac{s+\sigma}{(s+\sigma)^{2}+\omega_{d}^{2}}
Problem: Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :

Subproblem 0: $y(t)=e^{-a t}$


Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]

Final answer: The final answer is \frac{1}{s+a}. I hope it is correct.

Subproblem 1: $y(t)=e^{-\sigma t} \sin \omega_{d} t$
Answer: \frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}
Problem: Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\). 

Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input.
Answer: m\frac{dv}{dt}+bv=f
Problem: Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).

Subproblem 0: Formulate the state-determined equation of motion for the angular velocity $\omega$ as output and the torque $T$ as input.


Solution: The equation of motion is
\[
\boxed{I \frac{d \omega}{d t}+2 B \omega=T} \quad \text { or } \quad \frac{d \omega}{d t}=-\frac{2 B}{I} \omega+\frac{1}{I} T
\]

Final answer: The final answer is I \frac{d \omega}{d t}+2 B \omega=T. I hope it is correct.

Subproblem 1: Consider the case where:
\[
\begin{aligned}
I &=0.001 \mathrm{~kg}-\mathrm{m}^{2} \\
B &=0.005 \mathrm{~N}-\mathrm{m} / \mathrm{r} / \mathrm{s}
\end{aligned}
\]
What is the steady-state velocity \(\omega_{s s}\), in radians per second, when the input is a constant torque of 10 Newton-meters?
Answer: 1000
Problem: Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\). 

Subproblem 0: Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input.


Solution: The equation of motion is
\[
\boxed{m \frac{d v}{d t}+b v=f} \quad \text { or } \quad \frac{d v}{d t}=-\frac{b}{m} v+\frac{1}{m} f
\]

Final answer: The final answer is m \frac{d v}{d t}+b v=f. I hope it is correct.

Subproblem 1: Consider the case where:
\[
\begin{aligned}
m &=1000 \mathrm{~kg} \\
b &=100 \mathrm{~N} / \mathrm{m} / \mathrm{s}
\end{aligned}
\]
What is the steady-state velocity \(v_{s s}\) when the input is a constant force of 10 Newtons? Answer in meters per second.
Answer: 0.10
Problem: Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = -\frac{(4 s-10)}{s(s+2)(s+5)}$.
Use $u(t)$ to denote the unit step function.
Answer: (1-3e^{-2t}+2e^{-5t})u(t)
Problem: A signal has a Laplace transform
\[
X(s)=b+\frac{a}{s(s+a)}
\]
where \(a, b>0\), and with a region of convergence of \(|s|>0\). Find \(x(t), t>0\).
Answer: b\delta(t)+1-e^{-at}
Problem: Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :

\[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]
Answer: \frac{1}{b-a}(e^{-at}-e^{-bt})
Problem: Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).

Formulate the state-determined equation of motion for the angular velocity $\omega$ as output and the torque $T$ as input.
Answer: I\frac{d\omega}{dt}+2B\omega=T
Problem: Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = \frac{4}{s^2(s^2+4)}$.
Use $u(t)$ to denote the unit step function.
Answer: (t+\frac{1}{2}\sin{2t})u(t)
Problem: Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$.  We measure the voltage $v_{o}$ across the capacitor.  $v_{i}$ and $v_{o}$ share a ground reference.

Calculate the transfer function \(V_{o}(s) / V_{i}(s)\).
Answer: \frac{1}{LCs^{2}+RCs+1}
Problem: Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]

What is the time constant for this system?
Answer: 0.2
Problem: Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$.  We measure the voltage $v_{o}$ across the capacitor.  $v_{i}$ and $v_{o}$ share a ground reference.

Subproblem 0: Calculate the transfer function \(V_{o}(s) / V_{i}(s)\).


Solution: Using the voltage divider relationship:
\[
\begin{aligned}
V_{o}(s) &=\frac{Z_{e q}}{Z_{\text {total }}}V_{i}(s)=\frac{\frac{1}{C s}}{R+L s+\frac{1}{C s}} V_{i}(s) \\
\frac{V_{o}(s)}{V_{i}(s)} &=\boxed{\frac{1}{L C s^{2}+R C s+1}}
\end{aligned}
\]

Final answer: The final answer is \frac{1}{L C s^{2}+R C s+1}. I hope it is correct.

Subproblem 1: Let \(L=0.01 \mathrm{H}\). Choose the value of $C$ such that \(\omega_{n}=10^{5}\) and \(\zeta=0.05\).  Give your answer in Farads.
Answer: 1e-8
Problem: Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

What is the value of the natural frequency \(\omega_{n}\) in radians per second?
Answer: 100
Problem: Preamble: Consider a circuit in which a voltage source of voltage in $v_{i}(t)$ is connected in series with an inductor $L$ and capacitor $C$.  We consider the voltage across the capacitor $v_{o}(t)$ to be the output of the system.
Both $v_{i}(t)$ and $v_{o}(t)$ share ground reference.

Write the governing differential equation for this circuit.
Answer: \frac{d^{2}v_{o}}{dt^{2}}+\frac{v_{o}}{LC}=\frac{v_{i}}{LC}
Problem: Write (but don't solve) the equation of motion for a pendulum consisting of a mass $m$ attached to a rigid massless rod, suspended from the ceiling and free to rotate in a single vertical plane.  Let the rod (of length $l$) make an angle of $\theta$ with the vertical.  Gravity ($mg$) acts directly downward, the system input is a horizontal external force $f(t)$, and the system output is the angle $\theta(t)$.  
Note: Do NOT make the small-angle approximation in your equation.
Answer: ml\ddot{\theta}(t)-mg\sin\theta(t)=f(t)\cos\theta(t)
Problem: Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?


Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$

Final answer: The final answer is 100. I hope it is correct.

Subproblem 1: What is the value of the damping ratio \(\zeta\)? 


Solution: $\zeta=\frac{b}{2 \sqrt{k m}}$
So
$\zeta =\boxed{0.05}$

Final answer: The final answer is 0.05. I hope it is correct.

Subproblem 2: What is the value of the damped natural frequency \(\omega_{d}\) in radians per second? Give your answer to three significant figures.
Answer: 99.9
Problem: Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]

Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?


Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$

Final answer: The final answer is 100. I hope it is correct.

Subproblem 1: What is the value of the damping ratio \(\zeta\)?
Answer: 0.05
Problem: What is the speed of light in meters/second to 1 significant figure? Use the format $a \times 10^{b}$ where a and b are numbers.
Answer: 3e8
Problem: Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. 

Subproblem 0: Age of our universe when most He nuclei were formed in minutes: 


Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:


Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:


Solution: \boxed{10} Gyr.

Final answer: The final answer is 10. I hope it is correct.

Subproblem 3: Number of stars in our Galaxy: (Please format your answer as 'xen' representing $x * 10^n$)
Answer: 1e11
Problem: Preamble: In a parallel universe, the Boston baseball team made the playoffs.

Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate. Give your answer in terms of the speed of light, $c$.
Answer: \frac{1}{\sqrt{2}}c
Problem: Preamble: In the Sun, one of the processes in the He fusion chain is $p+p+e^{-} \rightarrow d+\nu$, where $d$ is a deuteron. Make the approximations that the deuteron rest mass is $2 m_{p}$, and that $m_{e} \approx 0$ and $m_{\nu} \approx 0$, since both the electron and the neutrino have negligible rest mass compared with the proton rest mass $m_{p}$.

In the lab frame, the two protons have the same energy $\gamma m_{p}$ and impact angle $\theta$, and the electron is at rest. Calculate the energy $E_{\nu}$ of the neutrino in the rest frame of the deuteron in terms of $\theta, m_{p}$ and $\gamma$.
Answer: m_{p}c^{2}(\gamma^{2}-1)\sin^{2}\theta
Problem: Preamble: In a parallel universe, the Boston baseball team made the playoffs.

Subproblem 0: Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume that he is still very close to home plate. Give your answer in terms of the speed of light, $c$.


Solution: Using the aberration formula with $\cos \theta^{\prime}=-1 / \sqrt{2}, \beta=1 / \sqrt{2}$, so $v=\boxed{\frac{1}{\sqrt{2}}c}$.

Final answer: The final answer is \frac{1}{\sqrt{2}}c. I hope it is correct.

Subproblem 1: A player standing on third base is wearing red socks emitting light of wavelength $\lambda_{\text {red}}$. What wavelength does Manny see in terms of $\lambda_{\text {red}}$?
Answer: \lambda_{\text{red}}/\sqrt{2}
Problem: Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. 

Subproblem 0: Age of our universe when most He nuclei were formed in minutes: 


Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:


Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:
Answer: 10
Problem: How many down quarks does a tritium ($H^3$) nucleus contain?
Answer: 5
Problem: How many up quarks does a tritium ($H^3$) nucleus contain?
Answer: 4
Problem: Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. 

Age of our universe when most He nuclei were formed in minutes:
Answer: 1
Problem: Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. 

Subproblem 0: Age of our universe when most He nuclei were formed in minutes: 


Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:


Solution: \boxed{400000} years.

Final answer: The final answer is 400000. I hope it is correct.

Subproblem 2: Age of our universe today in Gyr:


Solution: \boxed{10} Gyr.

Final answer: The final answer is 10. I hope it is correct.

Subproblem 3: Number of stars in our Galaxy: (Please format your answer as 'xen' representing $x * 10^n$)


Solution: \boxed{1e11}.

Final answer: The final answer is 1e11. I hope it is correct.

Subproblem 4: Light travel time to closest star (Sun!:) in minutes. (Please format your answer as an integer.)
Answer: 8
Problem: Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested. 

Subproblem 0: Age of our universe when most He nuclei were formed in minutes: 


Solution: \boxed{1} minute.

Final answer: The final answer is 1. I hope it is correct.

Subproblem 1: Age of our universe when hydrogen atoms formed in years:
Answer: 400000
Problem: Potassium metal can be used as the active surface in a photodiode because electrons are relatively easily removed from a potassium surface. The energy needed is $2.15 \times 10^{5} J$ per mole of electrons removed ( 1 mole $=6.02 \times 10^{23}$ electrons). What is the longest wavelength light (in nm) with quanta of sufficient energy to eject electrons from a potassium photodiode surface?
Answer: 560
Problem: Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places. 


Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\]

Final answer: The final answer is 4.4677. I hope it is correct.

Subproblem 1: the wave number $(\bar{v})$ in ${cm}^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 4 decimal places. 


Solution: $\bar{v}=\frac{1}{\lambda}=\frac{1}{6.7102 \times 10^{-7} m}=1.4903 \times 10^{6} m^{-1}= \boxed{1.4903e4} {cm}^{-1}$

Final answer: The final answer is 1.4903e4. I hope it is correct.

Subproblem 2: the wavelength $(\lambda)$ in nm, to 2 decimal places.
Answer: 671.02
Problem: What is the net charge of arginine in a solution of $\mathrm{pH} \mathrm{} 1.0$ ? Please format your answer as +n or -n.
Answer: +2
Problem: Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places. 


Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\]

Final answer: The final answer is 4.4677. I hope it is correct.

Subproblem 1: the wave number $(\bar{v})$ in ${cm}^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 4 decimal places.
Answer: 1.4903e4
Problem: Determine the atomic weight of ${He}^{++}$ in amu to 5 decimal places from the values of its constituents.
Answer: 4.03188
Problem: Preamble: Determine the following values from a standard radio dial. 

Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer. 


Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$

Final answer: The final answer is 188. I hope it is correct.

Subproblem 1: What is the maximum wavelength in m for broadcasts on the AM band? Format your answer as an integer.
Answer: 566
Problem: Determine the wavelength of radiation emitted by hydrogen atoms in angstroms upon electron transitions from $n=6$ to $n=2$.
Answer: 4100
Problem: Preamble: Determine the following values from a standard radio dial. 

Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer. 


Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$

Final answer: The final answer is 188. I hope it is correct.

Subproblem 1: What is the maximum wavelength in m for broadcasts on the AM band? Format your answer as an integer. 


Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
\[
\lambda_{\max }=\frac{3 \times 10^{8}}{530 \times 10^{3}}=\boxed{566} m
\]

Final answer: The final answer is 566. I hope it is correct.

Subproblem 2: What is the minimum wavelength in m (to 2 decimal places) for broadcasts on the FM band?
Answer: 2.78
Problem: Calculate the "Bohr radius" in angstroms to 3 decimal places for ${He}^{+}$.
Answer: 0.264
Problem: Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:

the frequency $(v)$ in Hz, to 4 decimal places.
Answer: 4.4677
Problem: Electromagnetic radiation of frequency $3.091 \times 10^{14} \mathrm{~Hz}$ illuminates a crystal of germanium (Ge). Calculate the wavelength of photoemission in meters generated by this interaction. Germanium is an elemental semiconductor with a band gap, $E_{g}$, of $0.7 \mathrm{eV}$. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places.
Answer: 1.77e-6
Problem: What is the energy gap (in eV, to 1 decimal place) between the electronic states $n=3$ and $n=8$ in a hydrogen atom?
Answer: 1.3
Problem: Determine for hydrogen the velocity in m/s of an electron in an ${n}=4$ state. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places.
Answer: 5.47e5
Problem: Preamble: A pure crystalline material (no impurities or dopants are present) appears red in transmitted light.

Subproblem 0: Is this material a conductor, semiconductor or insulator? Give the reasons for your answer.


Solution: If the material is pure (no impurity states present), then it must be classified as a \boxed{semiconductor} since it exhibits a finite "band gap" - i.e. to activate charge carriers, photons with energies in excess of "red" radiation are required.

Final answer: The final answer is semiconductor. I hope it is correct.

Subproblem 1: What is the approximate band gap $\left(\mathrm{E}_{g}\right)$ for this material in eV? Please round your answer to 1 decimal place.
Answer: 1.9