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Each of the two Magellan telescopes has a diameter of $6.5 \mathrm{~m}$. In one configuration the effective focal length is $72 \mathrm{~m}$. Find the diameter of the image of a planet (in $\mathrm{cm}$ ) at this focus if the angular diameter of the planet at the time of the observation is $45^{\prime \prime}$. | 1.6 |
A white dwarf star has an effective temperature, $T_{e}=50,000$ degrees Kelvin, but its radius, $R_{\mathrm{WD}}$, is comparable to that of the Earth. Take $R_{\mathrm{WD}}=10^{4} \mathrm{~km}\left(10^{7} \mathrm{~m}\right.$ or $\left.10^{9} \mathrm{~cm}\right)$. Compute the luminosity (power output) of the white dwarf. Treat the white dwarf as a blackbody radiator. Give your answer in units of ergs per second, to two significant figures. | 4.5e33 |
Preamble: A prism is constructed from glass and has sides that form a right triangle with the other two angles equal to $45^{\circ}$. The sides are $L, L$, and $H$, where $L$ is a leg and $H$ is the hypotenuse. A parallel light beam enters side $L$ normal to the surface, passes into the glass, and then strikes $H$ internally. The index of refraction of the glass is $n=1.5$.
Compute the critical angle for the light to be internally reflected at $H$. Give your answer in degrees to 3 significant figures. | 41.8 |
A particular star has an absolute magnitude $M=-7$. If this star is observed in a galaxy that is at a distance of $3 \mathrm{Mpc}$, what will its apparent magnitude be? | 20.39 |
Find the gravitational acceleration due to the Sun at the location of the Earth's orbit (i.e., at a distance of $1 \mathrm{AU}$ ). Give your answer in meters per second squared, and express it to one significant figure. | 0.006 |
Preamble: A collimated light beam propagating in water is incident on the surface (air/water interface) at an angle $\theta_w$ with respect to the surface normal.
Subproblem 0: If the index of refraction of water is $n=1.3$, find an expression for the angle of the light once it emerges from the water into the air, $\theta_a$, in terms of $\theta_w$.
Solution: Using Snell's law, $1.3 \sin{\theta_w} = \sin{\theta_a}$. So $\theta_a = \boxed{\arcsin{1.3 \sin{\theta_w}}}$.
Final answer: The final answer is \arcsin{1.3 \sin{\theta_w}}. I hope it is correct.
Subproblem 1: What is the critical angle, i.e., the critical value of $\theta_w$ such that the light will not emerge from the water? Leave your answer in terms of inverse trigonometric functions; i.e., do not evaluate the function. | np.arcsin(10/13) |
Find the theoretical limiting angular resolution (in arcsec) of a commercial 8-inch (diameter) optical telescope being used in the visible spectrum (at $\lambda=5000 \AA=500 \mathrm{~nm}=5 \times 10^{-5} \mathrm{~cm}=5 \times 10^{-7} \mathrm{~m}$). Answer in arcseconds to two significant figures. | 0.49 |
A star has a measured parallax of $0.01^{\prime \prime}$, that is, $0.01$ arcseconds. How far away is it, in parsecs? | 100 |
An extrasolar planet has been observed which passes in front of (i.e., transits) its parent star. If the planet is dark (i.e., contributes essentially no light of its own) and has a surface area that is $2 \%$ of that of its parent star, find the decrease in magnitude of the system during transits. | 0.022 |
If the Bohr energy levels scale as $Z^{2}$, where $Z$ is the atomic number of the atom (i.e., the charge on the nucleus), estimate the wavelength of a photon that results from a transition from $n=3$ to $n=2$ in Fe, which has $Z=26$. Assume that the Fe atom is completely stripped of all its electrons except for one. Give your answer in Angstroms, to two significant figures. | 9.6 |
If the Sun's absolute magnitude is $+5$, find the luminosity of a star of magnitude $0$ in ergs/s. A useful constant: the luminosity of the sun is $3.83 \times 10^{33}$ ergs/s. | 3.83e35 |
Preamble: A spectrum is taken of a single star (i.e., one not in a binary). Among the observed spectral lines is one from oxygen whose rest wavelength is $5007 \AA$. The Doppler shifted oxygen line from this star is observed to be at a wavelength of $5012 \AA$. The star is also observed to have a proper motion, $\mu$, of 1 arc second per year (which corresponds to $\sim 1.5 \times 10^{-13}$ radians per second of time). It is located at a distance of $60 \mathrm{pc}$ from the Earth. Take the speed of light to be $3 \times 10^8$ meters per second.
What is the component of the star's velocity parallel to its vector to the Earth (in kilometers per second)? | 300 |
The differential luminosity from a star, $\Delta L$, with an approximate blackbody spectrum, is given by:
\[
\Delta L=\frac{8 \pi^{2} c^{2} R^{2}}{\lambda^{5}\left[e^{h c /(\lambda k T)}-1\right]} \Delta \lambda
\]
where $R$ is the radius of the star, $T$ is its effective surface temperature, and $\lambda$ is the wavelength. $\Delta L$ is the power emitted by the star between wavelengths $\lambda$ and $\lambda+\Delta \lambda$ (assume $\Delta \lambda \ll \lambda)$. The star is at distance $d$. Find the star's spectral intensity $I(\lambda)$ at the Earth, where $I(\lambda)$ is defined as the power per unit area per unit wavelength interval. | \frac{2 \pi c^{2} R^{2}}{\lambda^{5}\left[e^{h c /(\lambda k T)}-1\right] d^{2}} |
Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \mathrm{kpc}$. The star has a temperature $T = 6 \times 10^{5} K$ and produces a flux of $10^{-12} \mathrm{erg} \cdot \mathrm{s}^{-1} \mathrm{cm}^{-2}$ at the Earth. Treat the star's surface as a blackbody radiator.
Subproblem 0: Find the luminosity of the star (in units of $\mathrm{erg} \cdot \mathrm{s}^{-1}$).
Solution: \[
L=4 \pi D^{2} \text { Flux }_{\text {Earth }}=10^{-12} 4 \pi\left(800 \times 3 \times 10^{21}\right)^{2}=\boxed{7e37} \mathrm{erg} \cdot \mathrm{s}^{-1}
\]
Final answer: The final answer is 7e37. I hope it is correct.
Subproblem 1: Compute the star's radius in centimeters. | 8.7e8 |
A star is at a distance from the Earth of $300 \mathrm{pc}$. Find its parallax angle, $\pi$, in arcseconds to one significant figure. | 0.003 |
The Sun's effective temperature, $T_{e}$, is 5800 Kelvin, and its radius is $7 \times 10^{10} \mathrm{~cm}\left(7 \times 10^{8}\right.$ m). Compute the luminosity (power output) of the Sun in erg/s. Treat the Sun as a blackbody radiator, and give your answer to one significant figure. | 4e33 |
Use the Bohr model of the atom to compute the wavelength of the transition from the $n=100$ to $n=99$ levels, in centimeters. [Uscful relation: the wavelength of $L \alpha$ ( $\mathrm{n}=2$ to $\mathrm{n}=1$ transition) is $1216 \AA$.] | 4.49 |
Preamble: A radio interferometer, operating at a wavelength of $1 \mathrm{~cm}$, consists of 100 small dishes, each $1 \mathrm{~m}$ in diameter, distributed randomly within a $1 \mathrm{~km}$ diameter circle.
What is the angular resolution of a single dish, in radians? | 0.01 |
Preamble: Orbital Dynamics: A binary system consists of two stars in circular orbit about a common center of mass, with an orbital period, $P_{\text {orb }}=10$ days. Star 1 is observed in the visible band, and Doppler measurements show that its orbital speed is $v_{1}=20 \mathrm{~km} \mathrm{~s}^{-1}$. Star 2 is an X-ray pulsar and its orbital radius about the center of mass is $r_{2}=3 \times 10^{12} \mathrm{~cm}=3 \times 10^{10} \mathrm{~m}$.
Subproblem 0: Find the orbital radius, $r_{1}$, of the optical star (Star 1) about the center of mass, in centimeters.
Solution: \[
\begin{gathered}
v_{1}=\frac{2 \pi r_{1}}{P_{\text {orb }}} \\
r_{1}=\frac{P_{\text {orb }} v_{1}}{2 \pi}=\boxed{2.75e11} \mathrm{~cm}
\end{gathered}
\]
Final answer: The final answer is 2.75e11. I hope it is correct.
Subproblem 1: What is the total orbital separation between the two stars, $r=r_{1}+r_{2}$ (in centimeters)? | 3.3e12 |
If a star cluster is made up of $10^{4}$ stars, each of whose absolute magnitude is $-5$, compute the combined apparent magnitude of the cluster if it is located at a distance of $1 \mathrm{Mpc}$. | 10 |
A galaxy moves directly away from us with a speed of $3000 \mathrm{~km} \mathrm{~s}^{-1}$. Find the wavelength of the $\mathrm{H} \alpha$ line observed at the Earth, in Angstroms. The rest wavelength of $\mathrm{H} \alpha$ is $6565 \AA$. Take the speed of light to be $3\times 10^8$ meters per second. | 6630 |
The Spitzer Space Telescope has an effective diameter of $85 \mathrm{cm}$, and a typical wavelength used for observation of $5 \mu \mathrm{m}$, or 5 microns. Based on this information, compute an estimate for the angular resolution of the Spitzer Space telescope in arcseconds. | 1.2 |
It has long been suspected that there is a massive black hole near the center of our Galaxy. Recently, a group of astronmers determined the parameters of a star that is orbiting the suspected black hole. The orbital period is 15 years, and the orbital radius is $0.12$ seconds of arc (as seen from the Earth). Take the distance to the Galactic center to be $8 \mathrm{kpc}$. Compute the mass of the black hole, starting from $F=m a$. Express your answer in units of the Sun's mass; i.e., answer the question `what is the ratio of masses between this black hole and our Sun'? Give your answer to 1 significant figure. (Assume that Newton's law of gravity is applicable for orbits sufficiently far from a black hole, and that the orbiting star satisfies this condition.) | 3e6 |
Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \mathrm{kpc}$. The star has a temperature $T = 6 \times 10^{5} K$ and produces a flux of $10^{-12} \mathrm{erg} \cdot \mathrm{s}^{-1} \mathrm{cm}^{-2}$ at the Earth. Treat the star's surface as a blackbody radiator.
Find the luminosity of the star (in units of $\mathrm{erg} \cdot \mathrm{s}^{-1}$). | 7e37 |
A large ground-based telescope has an effective focal length of 10 meters. Two astronomical objects are separated by 1 arc second in the sky. How far apart will the two corresponding images be in the focal plane, in microns? | 50 |
The equation of state for cold (non-relativistic) matter may be approximated as:
\[
P=a \rho^{5 / 3}-b \rho^{4 / 3}
\]
where $P$ is the pressure, $\rho$ the density, and $a$ and $b$ are fixed constants. Use a dimensional analysis of the equation of hydrostatic equilibrium to estimate the ``radius-mass'' relation for planets and low-mass white dwarfs whose material follows this equation of state. Specifically, find $R(M)$ in terms of $G$ and the constants $a$ and $b$. You should set all constants of order unity (e.g., $4, \pi, 3$, etc.) to $1.0$. [Hint: solve for $R(M)$ rather than $M(R)$ ]. You can check your answer by showing that for higher masses, $R \propto M^{-1 / 3}$, while for the lower-masses $R \propto M^{+1 / 3}$. | \frac{a M^{1 / 3}}{G M^{2 / 3}+b} |
Take the total energy (potential plus thermal) of the Sun to be given by the simple expression:
\[
E \simeq-\frac{G M^{2}}{R}
\]
where $M$ and $R$ are the mass and radius, respectively. Suppose that the energy generation in the Sun were suddenly turned off and the Sun began to slowly contract. During this contraction its mass, $M$, would remain constant and, to a fair approximation, its surface temperature would also remain constant at $\sim 5800 \mathrm{~K}$. Assume that the total energy of the Sun is always given by the above expression, even as $R$ gets smaller. By writing down a simple (differential) equation relating the power radiated at Sun's surface with the change in its total energy (using the above expression), integrate this equation to find the time (in years) for the Sun to shrink to $1 / 2$ its present radius. Answer in units of years. | 7.5e7 |
Preamble: Once a star like the Sun starts to ascend the giant branch its luminosity, to a good approximation, is given by:
\[
L=\frac{10^{5} L_{\odot}}{M_{\odot}^{6}} M_{\text {core }}^{6}
\]
where the symbol $\odot$ stands for the solar value, and $M_{\text {core }}$ is the mass of the He core of the star. Further, assume that as more hydrogen is burned to helium - and becomes added to the core - the conversion efficiency between rest mass and energy is:
\[
\Delta E=0.007 \Delta M_{\text {core }} c^{2} .
\]
Use these two expressions to write down a differential equation, in time, for $M_{\text {core }}$. For ease of writing, simply use the variable $M$ to stand for $M_{\text {core }}$. Leave your answer in terms of $c$, $M_{\odot}$, and $L_{\odot}$. | \frac{dM}{dt}=\frac{10^{5} L_{\odot}}{0.007 c^{2} M_{\odot}^{6}} M^{6} |
A star of radius, $R$, and mass, $M$, has an atmosphere that obeys a polytropic equation of state:
\[
P=K \rho^{5 / 3} \text {, }
\]
where $P$ is the gas pressure, $\rho$ is the gas density (mass per unit volume), and $K$ is a constant throughout the atmosphere. Assume that the atmosphere is sufficiently thin (compared to $R$ ) that the gravitational acceleration can be taken to be a constant.
Use the equation of hydrostatic equilibrium to derive the pressure as a function of height $z$ above the surface of the planet. Take the pressure at the surface to be $P_{0}$. | \left[P_{0}^{2 / 5}-\frac{2}{5} g K^{-3 / 5} z\right]^{5 / 2} |
An eclipsing binary consists of two stars of different radii and effective temperatures. Star 1 has radius $R_{1}$ and $T_{1}$, and Star 2 has $R_{2}=0.5 R_{1}$ and $T_{2}=2 T_{1}$. Find the change in bolometric magnitude of the binary, $\Delta m_{\text {bol }}$, when the smaller star is behind the larger star. (Consider only bolometric magnitudes so you don't have to worry about color differences.) | 1.75 |
Preamble: It has been suggested that our Galaxy has a spherically symmetric dark-matter halo with a density distribution, $\rho_{\text {dark }}(r)$, given by:
\[
\rho_{\text {dark }}(r)=\rho_{0}\left(\frac{r_{0}}{r}\right)^{2},
\]
where $\rho_{0}$ and $r_{0}$ are constants, and $r$ is the radial distance from the center of the galaxy. For star orbits far out in the halo you can ignore the gravitational contribution of the ordinary matter in the Galaxy.
Compute the rotation curve of the Galaxy (at large distances), i.e., find $v(r)$ for circular orbits. | \sqrt{4 \pi G \rho_{0} r_{0}^{2}} |
The Very Large Array (VLA) telescope has an effective diameter of $36 \mathrm{~km}$, and a typical wavelength used for observation at this facility might be $6 \mathrm{~cm}$. Based on this information, compute an estimate for the angular resolution of the VLA in arcseconds | 0.33 |
Subproblem 0: A particular star has an absolute magnitude $M=-7$. If this star is observed in a galaxy that is at a distance of $3 \mathrm{Mpc}$, what will its apparent magnitude be?
Solution: \[
\text { Given: } M=-7 \text { and } d=3 \mathrm{Mpc}
\]
\[
\begin{aligned}
& \text { Apparent Magnitude: } m=M+5 \log \left[\frac{d}{10 \mathrm{pc}}\right]=-7+5 \log \left[\frac{3 \times 10^{6}}{10}\right]=\boxed{20.39} \\
\end{aligned}
\]
Final answer: The final answer is 20.39. I hope it is correct.
Subproblem 1: What is the distance modulus to this galaxy? | 27.39 |
Find the distance modulus to the Andromeda galaxy (M31). Take the distance to Andromeda to be $750 \mathrm{kpc}$, and answer to three significant figures. | 24.4 |
The Hubble Space telescope has an effective diameter of $2.5 \mathrm{~m}$, and a typical wavelength used for observation by the Hubble might be $0.6 \mu \mathrm{m}$, or 600 nanometers (typical optical wavelength). Based on this information, compute an estimate for the angular resolution of the Hubble Space telescope in arcseconds. | 0.05 |
Preamble: A collimated light beam propagating in water is incident on the surface (air/water interface) at an angle $\theta_w$ with respect to the surface normal.
If the index of refraction of water is $n=1.3$, find an expression for the angle of the light once it emerges from the water into the air, $\theta_a$, in terms of $\theta_w$. | \arcsin{1.3 \sin{\theta_w}} |
What fraction of the rest mass energy is released (in the form of radiation) when a mass $\Delta M$ is dropped from infinity onto the surface of a neutron star with $M=1 M_{\odot}$ and $R=10$ $\mathrm{km}$ ? | 0.15 |
Preamble: The density of stars in a particular globular star cluster is $10^{6} \mathrm{pc}^{-3}$. Take the stars to have the same radius as the Sun, and to have an average speed of $10 \mathrm{~km} \mathrm{sec}^{-1}$.
Find the mean free path for collisions among stars. Express your answer in centimeters, to a single significant figure. | 2e27 |
For a gas supported by degenerate electron pressure, the pressure is given by:
\[
P=K \rho^{5 / 3}
\]
where $K$ is a constant and $\rho$ is the mass density. If a star is totally supported by degenerate electron pressure, use a dimensional analysis of the equation of hydrostatic equilibrium:
\[
\frac{d P}{d r}=-g \rho
\]
to determine how the radius of such a star depends on its mass, $M$. Specifically, you will find that $R$ is proportional to some power of $M$; what is that power? | -1./3 |
A galaxy moves directly away from us with speed $v$, and the wavelength of its $\mathrm{H} \alpha$ line is observed to be $6784 \AA$. The rest wavelength of $\mathrm{H} \alpha$ is $6565 \AA$. Find $v/c$. | 0.033 |
A candle has a power in the visual band of roughly $3$ Watts. When this candle is placed at a distance of $3 \mathrm{~km}$ it has the same apparent brightness as a certain star. Assume that this star has the same luminosity as the Sun in the visual band $\left(\sim 10^{26}\right.$ Watts $)$. How far away is the star (in pc)? | 0.5613 |
Preamble: A galaxy is found to have a rotation curve, $v(r)$, given by
\[
v(r)=\frac{\left(\frac{r}{r_{0}}\right)}{\left(1+\frac{r}{r_{0}}\right)^{3 / 2}} v_{0}
\]
where $r$ is the radial distance from the center of the galaxy, $r_{0}$ is a constant with the dimension of length, and $v_{0}$ is another constant with the dimension of speed. The rotation curve is defined as the orbital speed of test stars in circular orbit at radius $r$.
Find an expression for $\omega(r)$, where $\omega$ is the angular velocity. The constants $v_{0}$ and $r_{0}$ will appear in your answer. | \frac{v_{0}}{r_{0}} \frac{1}{\left(1+r / r_{0}\right)^{3 / 2}} |
Preamble: Orbital Dynamics: A binary system consists of two stars in circular orbit about a common center of mass, with an orbital period, $P_{\text {orb }}=10$ days. Star 1 is observed in the visible band, and Doppler measurements show that its orbital speed is $v_{1}=20 \mathrm{~km} \mathrm{~s}^{-1}$. Star 2 is an X-ray pulsar and its orbital radius about the center of mass is $r_{2}=3 \times 10^{12} \mathrm{~cm}=3 \times 10^{10} \mathrm{~m}$.
Find the orbital radius, $r_{1}$, of the optical star (Star 1) about the center of mass, in centimeters. | 2.75e11 |
Preamble: The density of stars in a particular globular star cluster is $10^{6} \mathrm{pc}^{-3}$. Take the stars to have the same radius as the Sun, and to have an average speed of $10 \mathrm{~km} \mathrm{sec}^{-1}$.
Subproblem 0: Find the mean free path for collisions among stars. Express your answer in centimeters, to a single significant figure.
Solution: \[
\begin{gathered}
\ell \simeq \frac{1}{n \sigma}=\frac{1}{10^{6} \mathrm{pc}^{-3} \pi R^{2}} \\
\ell \simeq \frac{1}{3 \times 10^{-50} \mathrm{~cm}^{-3} \times 1.5 \times 10^{22} \mathrm{~cm}^{2}} \simeq \boxed{2e27} \mathrm{~cm}
\end{gathered}
\]
Final answer: The final answer is 2e27. I hope it is correct.
Subproblem 1: Find the corresponding mean time between collisions. (Assume that the stars move in straight-line paths, i.e., are not deflected by gravitational interactions.) Answer in units of years, to a single significant figure. | 6e13 |
Preamble: A radio interferometer, operating at a wavelength of $1 \mathrm{~cm}$, consists of 100 small dishes, each $1 \mathrm{~m}$ in diameter, distributed randomly within a $1 \mathrm{~km}$ diameter circle.
Subproblem 0: What is the angular resolution of a single dish, in radians?
Solution: The angular resolution of a single dish is roughly given by the wavelength over its radius, in this case $\boxed{0.01}$ radians.
Final answer: The final answer is 0.01. I hope it is correct.
Subproblem 1: What is the angular resolution of the interferometer array for a source directly overhead, in radians? | 1e-5 |
If a star cluster is made up of $10^{6}$ stars whose absolute magnitude is the same as that of the Sun (+5), compute the combined magnitude of the cluster if it is located at a distance of $10 \mathrm{pc}$. | -10 |
A certain red giant has a radius that is 500 times that of the Sun, and a temperature that is $1 / 2$ that of the Sun's temperature. Find its bolometric (total) luminosity in units of the bolometric luminosity of the Sun. | 15625 |
Suppose air molecules have a collision cross section of $10^{-16} \mathrm{~cm}^{2}$. If the (number) density of air molecules is $10^{19} \mathrm{~cm}^{-3}$, what is the collision mean free path in cm? Answer to one significant figure. | 1e-3 |
Two stars have the same surface temperature. Star 1 has a radius that is $2.5$ times larger than the radius of star 2. Star 1 is ten times farther away than star 2. What is the absolute value of the difference in apparent magnitude between the two stars, rounded to the nearest integer? | 3 |
What is the slope of a $\log N(>F)$ vs. $\log F$ curve for a homogeneous distribution of objects, each of luminosity, $L$, where $F$ is the flux at the observer, and $N$ is the number of objects observed per square degree on the sky? | -3./2 |
Preamble: Comparison of Radio and Optical Telescopes.
The Very Large Array (VLA) is used to make an interferometric map of the Orion Nebula at a wavelength of $10 \mathrm{~cm}$. What is the best angular resolution of the radio image that can be produced, in radians? Note that the maximum separation of two antennae in the VLA is $36 \mathrm{~km}$. | 2.7778e-6 |
A globular cluster has $10^{6}$ stars each of apparent magnitude $+8$. What is the combined apparent magnitude of the entire cluster? | -7 |
Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \mathrm{kpc}$. The star has a temperature $T = 6 \times 10^{5} K$ and produces a flux of $10^{-12} \mathrm{erg} \cdot \mathrm{s}^{-1} \mathrm{cm}^{-2}$ at the Earth. Treat the star's surface as a blackbody radiator.
Subproblem 0: Find the luminosity of the star (in units of $\mathrm{erg} \cdot \mathrm{s}^{-1}$).
Solution: \[
L=4 \pi D^{2} \text { Flux }_{\text {Earth }}=10^{-12} 4 \pi\left(800 \times 3 \times 10^{21}\right)^{2}=\boxed{7e37} \mathrm{erg} \cdot \mathrm{s}^{-1}
\]
Final answer: The final answer is 7e37. I hope it is correct.
Subproblem 1: Compute the star's radius in centimeters.
Solution: \[
R=\left(L / 4 \pi \sigma T^{4}\right)^{1 / 2}=\boxed{8.7e8} \mathrm{~cm}=0.012 R_{\odot}
\]
Final answer: The final answer is 8.7e8. I hope it is correct.
Subproblem 2: At what wavelength is the peak of the emitted radiation? Answer in $\AA$. | 48 |
A Boolean function $F(A, B)$ is said to be universal if any arbitrary boolean function can be constructed by using nested $F(A, B)$ functions. A universal function is useful, since using it we can build any function we wish out of a single part. For example, when implementing boolean logic on a computer chip a universal function (called a 'gate' in logic-speak) can simplify design enormously. We would like to find a universal boolean function. In this problem we will denote the two boolean inputs $A$ and $B$ and the one boolean output as $C$.
First, to help us organize our thoughts, let's enumerate all of the functions we'd like to be able to construct. How many different possible one-output boolean functions of two variables are there? I.e., how many functions are there of the form $F(A, B)=C ?$ | 16 |
Unfortunately, a mutant gene can turn box people into triangles late in life. A laboratory test has been developed which can spot the gene early so that the dreaded triangle transformation can be prevented by medications. This test is 95 percent accurate at spotting the gene when it is there. However, the test gives a "false positive" $0.4$ percent of the time, falsely indicating that a healthy box person has the mutant gene. If $0.1$ percent (be careful - that's one-tenth of one percent) of the box people have the mutant gene, what's the probability that a box person actually has the mutant gene if the test indicates that he or she does? | 0.192 |
Buzz, the hot new dining spot on campus, emphasizes simplicity. It only has two items on the menu, burgers and zucchini. Customers make a choice as they enter (they are not allowed to order both), and inform the cooks in the back room by shouting out either "B" or "Z". Unfortunately the two letters sound similar so $8 \%$ of the time the cooks misinterpret what was said. The marketing experts who designed the restaurant guess that $90 \%$ of the orders will be for burgers and $10 \%$ for zucchini.
The cooks can hear one order per second. The customers arrive at the rate of one per second. One of the chefs says that this system will never work because customers can only send one bit per second, the rate at which orders can be accepted, so you could barely keep up even if there were no noise in the channel. You are hired as an outside consultant to deal with the problem.
What is the channel capacity $\mathrm{C}$ of this communication channel in bits per second? | 0.5978 |
Preamble: Given the following data from an Experimental Forest, answer the following questions. Show your work and units.
$\begin{array}{ll}\text { Total vegetative biomass } & 80,000 \mathrm{kcal} \mathrm{m}^{-2} \\ \text { Detritus and organic matter in soil } & 120,000 \mathrm{kcal } \mathrm{m}^{-2} \\ \text { Total Gross Primary Productivity } & 20,000 \mathrm{kcal } \mathrm{m}^{-2} \mathrm{yr}^{-1} \\ \text { Total Plant Respiration } & 5,000 \mathrm{kcal} \mathrm{m}^{-2} \mathrm{yr}^{-1} \\ \text { Total Community Respiration } & 9,000 \mathrm{kcal} \mathrm{m}^{-2} \mathrm{yr}^{-1}\end{array}$
Subproblem 0: What is the net primary productivity of the forest?
Solution: NPP $=$ GPP $-R_{A}=20,000-5,000=\boxed{15000} \mathrm{kcal} \mathrm{m}^{-2} \mathrm{yr}^{-1}$
Final answer: The final answer is 15000. I hope it is correct.
Subproblem 1: What is the net community production? | 11000 |
Preamble: A population of 100 ferrets is introduced to a large island in the beginning of 1990 . Ferrets have an intrinsic growth rate, $r_{\max }$ of $1.3 \mathrm{yr}^{-1}$.
Subproblem 0: Assuming unlimited resources-i.e., there are enough resources on this island to last the ferrets for hundreds of years-how many ferrets will there be on the island in the year 2000? (Show your work!)
Solution: $N_o = 100$ (in 1990)
\\
$N = ?$ (in 2000)
\\
$t = 10$ yr
\\
$r = 1.3 \text{yr}^{-1}$
\\
$N = N_{o}e^{rt} = 100*e^{(1.3/\text{yr})(10 \text{yr})} = 4.4 x 10^7$ ferrets
\\
There will be \boxed{4.4e7} ferrets on the island in the year 2000.
Final answer: The final answer is 4.4e7. I hope it is correct.
Subproblem 1: What is the doubling time of the ferret population? (Show your work!) | 0.53 |
Preamble: Given the following data from an Experimental Forest, answer the following questions. Show your work and units.
$\begin{array}{ll}\text { Total vegetative biomass } & 80,000 \mathrm{kcal} \mathrm{m}^{-2} \\ \text { Detritus and organic matter in soil } & 120,000 \mathrm{kcal } \mathrm{m}^{-2} \\ \text { Total Gross Primary Productivity } & 20,000 \mathrm{kcal } \mathrm{m}^{-2} \mathrm{yr}^{-1} \\ \text { Total Plant Respiration } & 5,000 \mathrm{kcal} \mathrm{m}^{-2} \mathrm{yr}^{-1} \\ \text { Total Community Respiration } & 9,000 \mathrm{kcal} \mathrm{m}^{-2} \mathrm{yr}^{-1}\end{array}$
What is the net primary productivity of the forest? | 15000 |
Preamble: The Peak District Moorlands in the United Kingdom store 20 million tonnes of carbon, almost half of the carbon stored in the soils of the entire United Kingdom (the Moorlands are only $8 \%$ of the land area). In pristine condition, these peatlands can store an additional 13,000 tonnes of carbon per year.
Given this rate of productivity, how long did it take for the Peatlands to sequester this much carbon? | 1538 |
Preamble: A population of 100 ferrets is introduced to a large island in the beginning of 1990 . Ferrets have an intrinsic growth rate, $r_{\max }$ of $1.3 \mathrm{yr}^{-1}$.
Assuming unlimited resources-i.e., there are enough resources on this island to last the ferrets for hundreds of years-how many ferrets will there be on the island in the year 2000? (Show your work!) | 4.4e7 |
Preamble: The following subproblems refer to a circuit with the following parameters. Denote by $I(t)$ the current (where the positive direction is, say, clockwise) in the circuit and by $V(t)$ the voltage increase across the voltage source, at time $t$. Denote by $R$ the resistance of the resistor and $C$ the capacitance of the capacitor (in units which we will not specify)-both positive numbers. Then
\[
R \dot{I}+\frac{1}{C} I=\dot{V}
\]
Subproblem 0: Suppose that $V$ is constant, $V(t)=V_{0}$. Solve for $I(t)$, with initial condition $I(0)$.
Solution: When $V$ is constant, the equation becomes $R \dot{I}+\frac{1}{C} I=0$, which is separable. Solving gives us
\[
I(t)=\boxed{I(0) e^{-\frac{t}{R C}}
}\].
Final answer: The final answer is I(0) e^{-\frac{t}{R C}}
. I hope it is correct.
Subproblem 1: It is common to write the solution to the previous subproblem in the form $c e^{-t / \tau}$. What is $c$ in this case? | I(0) |
Consider the following "mixing problem." A tank holds $V$ liters of salt water. Suppose that a saline solution with concentration of $c \mathrm{gm} /$ liter is added at the rate of $r$ liters/minute. A mixer keeps the salt essentially uniformly distributed in the tank. A pipe lets solution out of the tank at the same rate of $r$ liters/minute. The differential equation for the amount of salt in the tank is given by
\[
x^{\prime}+\frac{r}{V} x-r c=0 .
\]
Suppose that the out-flow from this tank leads into another tank, also of volume 1 , and that at time $t=1$ the water in it has no salt in it. Again there is a mixer and an outflow. Write down a differential equation for the amount of salt in this second tank, as a function of time, assuming the amount of salt in the second tank at moment $t$ is given by $y(t)$, and the amount of salt in the first tank at moment $t$ is given by $x(t)$. | y^{\prime}+r y-r x(t)=0 |
Find the general solution of $x^{2} y^{\prime}+2 x y=\sin (2 x)$, solving for $y$. Note that a general solution to a differential equation has the form $x=x_{p}+c x_{h}$ where $x_{h}$ is a nonzero solution of the homogeneous equation $\dot{x}+p x=0$. Additionally, note that the left hand side is the derivative of a product. | c x^{-2}-\frac{\cos (2 x)}{2 x^{2}} |
An African government is trying to come up with good policy regarding the hunting of oryx. They are using the following model: the oryx population has a natural growth rate of $k$, and we suppose a constant harvesting rate of $a$ oryxes per year.
Write down an ordinary differential equation describing the evolution of the oryx population given the dynamics above, using $x(t)$ to denote the oryx population (the number of individual oryx(es)) at time $t$, measured in years. | \frac{d x}{d t}=k x-a |
If the complex number $z$ is given by $z = 1+\sqrt{3} i$, what is the magnitude of $z^2$? | 4 |
In the polar representation $(r, \theta)$ of the complex number $z=1+\sqrt{3} i$, what is $r$? | 2 |
Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$.
Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers. | 1+\sqrt{3} i |
Subproblem 0: Find the general solution of the differential equation $y^{\prime}=x-2 y$ analytically using integrating factors, solving for $y$. Note that a function $u(t)$ such that $u \dot{x}+u p x=\frac{d}{d t}(u x)$ is an integrating factor. Additionally, note that a general solution to a differential equation has the form $x=x_{p}+c x_{h}$ where $x_{h}$ is a nonzero solution of the homogeneous equation $\dot{x}+p x=0$.
Solution: In standard form, $y^{\prime}+2 y=x$, so $u=C e^{2 x}$. Then $y=u^{-1} \int u x d x=e^{-2 x} \int x e^{2 x} d x$. Integrating by parts yields $\int x e^{2 x} d x=$ $\frac{x}{2} e^{2 x}-\frac{1}{2} \int e^{2 x} d x=\frac{x}{2} e^{2 x}-\frac{1}{4} e^{2 x}+c$. Therefore, $y=\boxed{x / 2-1 / 4+c e^{-2 x}}$.
Final answer: The final answer is x / 2-1 / 4+c e^{-2 x}. I hope it is correct.
Subproblem 1: For what value of $c$ does the straight line solution occur? | 0 |
Preamble: The following subproblems relate to applying Euler's Method (a first-order numerical procedure for solving ordinary differential equations with a given initial value) onto $y^{\prime}=y^{2}-x^{2}=F(x, y)$ at $y(0)=-1$, with $h=0.5$. Recall the notation \[x_{0}=0, y_{0}=-1, x_{n+1}=x_{h}+h, y_{n+1}=y_{n}+m_{n} h, m_{n}=F\left(x_{n}, y_{n}\right)\].
Use Euler's method to estimate the value at $x=1.5$. | -0.875 |
Rewrite the function $f(t) = \cos (2 t)+\sin (2 t)$ in the form $A \cos (\omega t-\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$. | \sqrt{2} \cos (2 t-\pi / 4) |
Given the ordinary differential equation $\ddot{x}-a^{2} x=0$, where $a$ is a nonzero real-valued constant, find a solution $x(t)$ to this equation such that $x(0) = 0$ and $\dot{x}(0)=1$. | \frac{1}{2a}(\exp{a*t} - \exp{-a*t}) |
Find a solution to the differential equation $\ddot{x}+\omega^{2} x=0$ satisfying the initial conditions $x(0)=x_{0}$ and $\dot{x}(0)=\dot{x}_{0}$. | x_{0} \cos (\omega t)+$ $\dot{x}_{0} \sin (\omega t) / \omega |
Find the complex number $a+b i$ with the smallest possible positive $b$ such that $e^{a+b i}=1+\sqrt{3} i$. | \ln 2 + i\pi / 3 |
Subproblem 0: Find the general solution of the differential equation $\dot{x}+2 x=e^{t}$, using $c$ for the arbitrary constant of integration which will occur.
Solution: We can use integrating factors to get $(u x)^{\prime}=u e^{t}$ for $u=e^{2 t}$. Integrating yields $e^{2 t} x=e^{3 t} / 3+c$, or $x=\boxed{\frac{e^{t}} {3}+c e^{-2 t}}$.
Final answer: The final answer is \frac{e^{t}} {3}+c e^{-2 t}. I hope it is correct.
Subproblem 1: Find a solution of the differential equation $\dot{x}+2 x=e^{t}$ of the form $w e^{t}$, where $w$ is a constant (which you should find). | e^{t} / 3 |
Subproblem 0: For $\omega \geq 0$, find $A$ such that $A \cos (\omega t)$ is a solution of $\ddot{x}+4 x=\cos (\omega t)$.
Solution: If $x=A \cos (\omega t)$, then taking derivatives gives us $\ddot{x}=-\omega^{2} A \cos (\omega t)$, and $\ddot{x}+4 x=\left(4-\omega^{2}\right) A \cos (\omega t)$. Then $A=\boxed{\frac{1}{4-\omega^{2}}}$.
Final answer: The final answer is \frac{1}{4-\omega^{2}}. I hope it is correct.
Subproblem 1: For what value of $\omega$ does resonance occur? | 2 |
Subproblem 0: Find a purely sinusoidal solution of $\frac{d^{4} x}{d t^{4}}-x=\cos (2 t)$.
Solution: We choose an exponential input function whose real part is $\cos (2 t)$, namely $e^{2 i t}$. Since $p(s)=s^{4}-1$ and $p(2 i)=15 \neq 0$, the exponential response formula yields the solution $\frac{e^{2 i t}}{15}$. A sinusoidal solution to the original equation is given by the real part: $\boxed{\frac{\cos (2 t)}{15}}$.
Final answer: The final answer is \frac{\cos (2 t)}{15}. I hope it is correct.
Subproblem 1: Find the general solution to $\frac{d^{4} x}{d t^{4}}-x=\cos (2 t)$, denoting constants as $C_{1}, C_{2}, C_{3}, C_{4}$. | \frac{\cos (2 t)}{15}+C_{1} e^{t}+C_{2} e^{-t}+C_{3} \cos (t)+C_{4} \sin (t) |
For $\omega \geq 0$, find $A$ such that $A \cos (\omega t)$ is a solution of $\ddot{x}+4 x=\cos (\omega t)$. | \frac{1}{4-\omega^{2}} |
Find a solution to $\dot{x}+2 x=\cos (2 t)$ in the form $k_0\left[f(k_1t) + g(k_2t)\right]$, where $f, g$ are trigonometric functions. Do not include homogeneous solutions to this ODE in your solution. | \frac{\cos (2 t)+\sin (2 t)}{4} |
Preamble: The following subproblems refer to the differential equation. $\ddot{x}+4 x=\sin (3 t)$
Find $A$ so that $A \sin (3 t)$ is a solution of $\ddot{x}+4 x=\sin (3 t)$. | -0.2 |
Find the general solution of the differential equation $y^{\prime}=x-2 y$ analytically using integrating factors, solving for $y$. Note that a function $u(t)$ such that $u \dot{x}+u p x=\frac{d}{d t}(u x)$ is an integrating factor. Additionally, note that a general solution to a differential equation has the form $x=x_{p}+c x_{h}$ where $x_{h}$ is a nonzero solution of the homogeneous equation $\dot{x}+p x=0$. | x / 2-1 / 4+c e^{-2 x} |
Subproblem 0: Find a purely exponential solution of $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$.
Solution: The characteristic polynomial of the homogeneous equation is given by $p(s)=$ $s^{4}-1$. Since $p(-2)=15 \neq 0$, the exponential response formula gives the solution $\frac{e^{-2 t}}{p(-2)}=\boxed{\frac{e^{-2 t}}{15}}$.
Final answer: The final answer is \frac{e^{-2 t}}{15}. I hope it is correct.
Subproblem 1: Find the general solution to $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$, denoting constants as $C_{1}, C_{2}, C_{3}, C_{4}$. | \frac{e^{-2 t}}{15}+C_{1} e^{t}+C_{2} e^{-t}+ C_{3} \cos (t)+C_{4} \sin (t) |
Preamble: Consider the differential equation $\ddot{x}+\omega^{2} x=0$. \\
A differential equation $m \ddot{x}+b \dot{x}+k x=0$ (where $m, b$, and $k$ are real constants, and $m \neq 0$ ) has corresponding characteristic polynomial $p(s)=m s^{2}+b s+k$.\\
What is the characteristic polynomial $p(s)$ of $\ddot{x}+\omega^{2} x=0$? | s^{2}+\omega^{2} |
Rewrite the function $\cos (\pi t)-\sqrt{3} \sin (\pi t)$ in the form $A \cos (\omega t-\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$. | 2 \cos (\pi t+\pi / 3) |
Preamble: The following subproblems refer to the damped sinusoid $x(t)=A e^{-a t} \cos (\omega t)$.
What is the spacing between successive maxima of $x(t)$? Assume that $\omega \neq 0$. | 2 \pi / \omega |
Preamble: The following subproblems refer to a spring/mass/dashpot system driven through the spring modeled by the equation $m \ddot{x}+b \dot{x}+k x=k y$. Here $x$ measures the position of the mass, $y$ measures the position of the other end of the spring, and $x=y$ when the spring is relaxed.
In this system, regard $y(t)$ as the input signal and $x(t)$ as the system response. Take $m=1, b=3, k=4, y(t)=A \cos t$. Replace the input signal by a complex exponential $y_{c x}(t)$ of which it is the real part, and compute the exponential ("steady state") system response $z_p(t)$; leave your answer in terms of complex exponentials, i.e. do not take the real part. | \frac{4 A}{3+3 i} e^{i t} |
Preamble: The following subproblems refer to a circuit with the following parameters. Denote by $I(t)$ the current (where the positive direction is, say, clockwise) in the circuit and by $V(t)$ the voltage increase across the voltage source, at time $t$. Denote by $R$ the resistance of the resistor and $C$ the capacitance of the capacitor (in units which we will not specify)-both positive numbers. Then
\[
R \dot{I}+\frac{1}{C} I=\dot{V}
\]
Suppose that $V$ is constant, $V(t)=V_{0}$. Solve for $I(t)$, with initial condition $I(0)$. | I(0) e^{-\frac{t}{R C}} |
Subproblem 0: 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.
Solution: Using integrating factors, we get $e^{2 t} z=e^{(2+2 i) t} /(2+2 i)+C$, or $z=\boxed{\frac{e^{2 i t}}{(2+2 i)}+C e^{-2 t}}$, where $C$ is any complex number.
Final answer: The final answer is \frac{e^{2 i t}}{(2+2 i)}+C e^{-2 t}. I hope it is correct.
Subproblem 1: Find a solution of the differential equation $\dot{z}+2 z=e^{2 i t}$ in the form $w e^{t}$, where $w$ is a constant (which you should find). | \frac{e^{2 i t}}{(2+2 i)} |
Preamble: The following subproblems consider a second order mass/spring/dashpot system driven by a force $F_{\text {ext }}$ acting directly on the mass: $m \ddot{x}+b \dot{x}+k x=F_{\text {ext }}$. So the input signal is $F_{\text {ext }}$ and the system response is $x$. We're interested in sinusoidal input signal, $F_{\text {ext }}(t)=A \cos (\omega t)$, and in the steady state, sinusoidal system response, $x_{p}(t)=g A \cos (\omega t-\phi)$. Here $g$ is the gain of the system and $\phi$ is the phase lag. Both depend upon $\omega$, and we will consider how that is the case. \\
Take $A=1$, so the amplitude of the system response equals the gain, and take $m=1, b=\frac{1}{4}$, and $k=2$.\\
Compute the complex gain $H(\omega)$ of this system. (This means: make the complex replacement $F_{\mathrm{cx}}=e^{i \omega t}$, and express the exponential system response $z_{p}$ as a complex multiple of $F_{\mathrm{cx}}, i.e. z_{p}=H(\omega) F_{\mathrm{cx}}$). | \frac{2-\omega^{2}-\omega i / 4}{\omega^{4}-\frac{63}{16} \omega^{2}+4} |
Preamble: The following subproblems refer to the following "mixing problem": A tank holds $V$ liters of salt water. Suppose that a saline solution with concentration of $c \mathrm{gm} /$ liter is added at the rate of $r$ liters/minute. A mixer keeps the salt essentially uniformly distributed in the tank. A pipe lets solution out of the tank at the same rate of $r$ liters/minute.
Write down the differential equation for the amount of salt in the tank in standard linear form. [Not the concentration!] Use the notation $x(t)$ for the number of grams of salt in the tank at time $t$. | x^{\prime}+\frac{r}{V} x-r c=0 |
Find the polynomial solution of $\ddot{x}-x=t^{2}+t+1$, solving for $x(t)$. | -t^2 - t - 3 |
Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$.
Subproblem 0: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers.
Solution: Using Euler's formula, we find that the answer is $\boxed{1+\sqrt{3} i}$.
Final answer: The final answer is 1+\sqrt{3} i. I hope it is correct.
Subproblem 1: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{2(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers.
Solution: $e^{n(a+b i)}=(1+\sqrt{3} i)^{n}$, so the answer is $\boxed{-2+2 \sqrt{3} i}$.
Final answer: The final answer is -2+2 \sqrt{3} i. I hope it is correct.
Subproblem 2: Rewrite $e^{3(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers. | -8 |
Find a purely sinusoidal solution of $\frac{d^{4} x}{d t^{4}}-x=\cos (2 t)$. | \frac{\cos (2 t)}{15} |
Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$.
Subproblem 0: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers.
Solution: Using Euler's formula, we find that the answer is $\boxed{1+\sqrt{3} i}$.
Final answer: The final answer is 1+\sqrt{3} i. I hope it is correct.
Subproblem 1: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{2(a+b i)}$ in the form $x + yi$, where $x, y$ are real numbers. | -2+2 \sqrt{3} i |
Find a solution of $\ddot{x}+4 x=\cos (2 t)$, solving for $x(t)$, by using the ERF on a complex replacement. The ERF (Exponential Response Formula) states that a solution to $p(D) x=A e^{r t}$ is given by $x_{p}=A \frac{e^{r t}}{p(r)}$, as long as $\left.p (r\right) \neq 0$). The ERF with resonance assumes that $p(r)=0$ and states that a solution to $p(D) x=A e^{r t}$ is given by $x_{p}=A \frac{t e^{r t}}{p^{\prime}(r)}$, as long as $\left.p^{\prime} ( r\right) \neq 0$. | \frac{t}{4} \sin (2 t) |
Given the ordinary differential equation $\ddot{x}-a^{2} x=0$, where $a$ is a nonzero real-valued constant, find a solution $x(t)$ to this equation such that $x(0) = 1$ and $\dot{x}(0)=0$. | \frac{1}{2}(\exp{a*t} + \exp{-a*t}) |
Find the general solution of the differential equation $\dot{x}+2 x=e^{t}$, using $c$ for the arbitrary constant of integration which will occur. | \frac{e^{t}} {3}+c e^{-2 t} |
Find a solution of $\ddot{x}+3 \dot{x}+2 x=t e^{-t}$ in the form $x(t)=u(t) e^{-t}$ for some function $u(t)$. Use $C$ for an arbitrary constant, should it arise. | \left(\frac{t^{2}}{2}-t+C\right) e^{-t} |
If the complex number $z$ is given by $z = 1+\sqrt{3} i$, what is the real part of $z^2$? | -2 |
Find a purely exponential solution of $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$. | \frac{e^{-2 t}}{15} |
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