question stringlengths 58 998 | answer stringlengths 1 210 | metadata dict |
|---|---|---|
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Start with:\n\\[\ns=\\alpha f \\text {, }\n\\]\nwhere $s$ is the diameter of the image, $f$ the focal length, and $\\alpha$ the angular diameter of the planet. For the values given in the problem:\n\\[\ns=\\frac{45}{3600} \\frac{\\pi}{180... |
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... | 4.5e33 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{aligned}\nL=4 \\pi R^{2} \\sigma T_{e}^{4} &=4 \\pi\\left(10^{9}\\right)^{2}\\left(5.7 \\times 10^{-5}\\right)(50,000)^{4} \\operatorname{ergs~s}^{-1} \\\\\nL & \\simeq \\boxed{4.5e33} \\mathrm{ergs} \\mathrm{s}^{-1} \\simeq ... |
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 $H$ inte... | 41.8 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "From Snell's law we have:\n\\[\n\\begin{gathered}\nn_{g} \\sin \\left(\\theta_{g}\\right)=n_{\\text {air }} \\sin \\left(\\theta_{\\text {air }}\\right) \\\\\n\\sin \\left(\\theta_{\\text {crit }}\\right)=\\frac{1}{1.5} \\sin \\left(90^{\... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\text { Given: } M=-7 \\text { and } d=3 \\mathrm{Mpc}\n\\]\n\\[\n\\begin{aligned}\n & \\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]=\\bo... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\begin{equation}\nF = ma = \\frac{GM_{\\odot}m}{r^2},\n\\end{equation}\nso \n\\begin{equation}\na = \\frac{GM_{\\odot}{r^2}}\n\\end{equation}\n\nPlugging in values for $G$, $M_{\\odot}$, and $r$ gives $a = \\boxed{0.006}$ meters per seco... |
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, $\t... | np.arcsin(10/13) | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The relation derived in the previous problem is $\\theta_a = \\arcsin{1.3 \\sin{\\theta_w}}$. The critical angle thus occurs when $1.3 \\sin{\\theta_w}$ exceeds unity, because then there is no corresponding solution for $\\theta_a$. So ... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\theta=1.22 \\frac{\\lambda}{D}=1.22 \\frac{5 \\times 10^{-5} \\mathrm{~cm}}{8 \\times 2.54 \\mathrm{~cm}}=2.46 \\times 10^{-6} \\text { radians }=\\boxed{0.49} \\operatorname{arcsecs}\n\\]",
"type": "Introduction to Astronomy (8.... |
A star has a measured parallax of $0.01^{\prime \prime}$, that is, $0.01$ arcseconds. How far away is it, in parsecs? | 100 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Almost by definition, it is $\\boxed{100}$ parsecs away.",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The flux goes from a maximum of $F_{0}$, when the planet is not blocking any light, to $0.98 F_{0}$ when the planet is in front of the stellar disk. So, the uneclipsed magnitude is:\n\\[\nm_{0}=-2.5 \\log \\left(F_{0} / F_{\\text {ref }}\... |
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. G... | 9.6 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{gathered}\nh \\nu=13.6 Z^{2}\\left[\\frac{1}{n_{f}^{2}}-\\frac{1}{n_{i}^{2}}\\right] \\mathrm{eV} \\\\\nh \\nu=13.6 \\times 26^{2}\\left[\\frac{1}{2^{2}}-\\frac{1}{3^{2}}\\right] \\mathrm{eV} \\\\\nh \\nu=1280 \\mathrm{eV}=1.... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The relation between luminosity and absolute magnitude is: $m - n = 2.5 \\log (f_n/f_m)$; note the numerator and denominator: brighter objects have numericallly smaller magnitudes. If a star has magnitude $0$, then since the difference i... |
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, $\mu$, ... | 300 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "To find this longitudinal velocity component, we use the Doppler shift, finding $V_{r}=\\frac{\\Delta \\lambda}{\\lambda} c=\\frac{5}{5000} c=\\boxed{300} \\mathrm{~km} / \\mathrm{s}$.",
"type": "Introduction to Astronomy (8.282J Spring... |
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 wavel... | \frac{2 \pi c^{2} R^{2}}{\lambda^{5}\left[e^{h c /(\lambda k T)}-1\right] d^{2}} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\nI(\\lambda)=\\frac{1}{4 \\pi d^{2}} \\frac{\\Delta L}{\\Delta \\lambda}=\\boxed{\\frac{2 \\pi c^{2} R^{2}}{\\lambda^{5}\\left[e^{h c /(\\lambda k T)}-1\\right] d^{2}}}\n\\]",
"type": "Introduction to Astronomy (8.282J Spring 2006)"... |
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: Fin... | 8.7e8 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n R=\\left(L / 4 \\pi \\sigma T^{4}\\right)^{1 / 2}=\\boxed{8.7e8} \\mathrm{~cm}=0.012 R_{\\odot}\n\\]",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{aligned}\nD &=1 \\mathrm{pc} / \\pi^{\\prime \\prime} \\\\\n\\pi^{\\prime \\prime} &=1 \\mathrm{pc} / 300 \\mathrm{pc} \\\\\n\\pi^{\\prime \\prime} &=\\boxed{0.003}^{\\prime \\prime}\n\\end{aligned}\n\\]",
"type": "Introduc... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Using the standard formula for power output of a blackbody radiator gives $P = \\sigma A T^4$, where the area in this case is $4\\piR_{sun}^2$. Plugging in the numbers given in the problem yields that the sun's power output is (to one si... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The inverse wavelength of radiation is proportional to the energy difference between the initial and final energy levels. So for our transition of interest, we have \n\\begin{equation}\n \\lambda^{-1} = R(\\frac{1}{99^2} - \\frac{1}{100... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The angular resolution of a single dish is roughly given by the wavelength over its radius, in this case $\\boxed{0.01}$ radians.",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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-... | 3.3e12 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n r=r_{1}+r_{2}=2.75 \\times 10^{11}+3 \\times 10^{12}=\\boxed{3.3e12} \\quad \\mathrm{~cm}\n\\]",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The absolute magnitude of one of the stars is given by:\n\\[\nM=-2.5 \\log \\left(L / L_{\\mathrm{ref}}\\right)=-5\n\\]\nwhere $L$ is the stellar luminosity, and $L_{\\text {ref }}$ is the luminosity of a zero magnitude star. This equatio... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "We have that the velocity of the galaxy is $0.01$ times $c$, the speed of light. So, using Doppler effect formulas,\n\\begin{equation}\n\\lambda_{obs} = (6565 \\AA)(1 + v/c) = (6565 \\AA)(1.01)\n\\end{equation}\nSo the answer is $\\boxed... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Using the formula for angular resolution $\\theta$ in terms of the effective size $d$ and the wavelength $\\lambda$, namely $\\theta = \\lambda/d$, gives \\boxed{1.2} arcseconds.",
"type": "Introduction to Astronomy (8.282J Spring 2006)... |
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 d... | 3e6 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The force of gravitational attraction between the black hole (of mass $M_{BH}$) and the star (of mass $M_s$) is given by\n\\begin{equation}\nF = \\frac{G M_{BH} M_s}{R^2},\n\\end{equation}\nwhere $R$ is the distance between the star and b... |
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 luminosi... | 7e37 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n 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}\n\\]",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\ns=f \\theta=1000 \\mathrm{~cm} \\times \\frac{1}{2 \\times 10^{5}} \\text { radians }=0.005 \\mathrm{~cm}=\\boxed{50} \\mu \\mathrm{m}\n\\]",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 pl... | \frac{a M^{1 / 3}}{G M^{2 / 3}+b} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{gathered}\n\\frac{d P}{d r}=-g \\rho \\\\\n\\frac{a \\rho^{5 / 3}-b \\rho^{4 / 3}}{R} \\sim\\left(\\frac{G M}{R^{2}}\\right)\\left(\\frac{M}{R^{3}}\\right) \\\\\n\\frac{a M^{5 / 3}}{R^{6}}-\\frac{b M^{4 / 3}}{R^{5}} \\sim\\le... |
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 i... | 7.5e7 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{gathered}\nL=4 \\pi \\sigma R^{2} T^{4}=d E / d t=\\left(\\frac{G M^{2}}{R^{2}}\\right) \\frac{d R}{d t} \\\\\n\\int_{R}^{0.5 R} \\frac{d R}{R^{4}}=-\\int_{0}^{t} \\frac{4 \\pi \\sigma T^{4}}{G M^{2}} d t \\\\\n-\\frac{1}{3(R... |
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, as... | \frac{dM}{dt}=\frac{10^{5} L_{\odot}}{0.007 c^{2} M_{\odot}^{6}} M^{6} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\nL \\equiv \\frac{\\Delta E}{\\Delta t}=\\frac{0.007 \\Delta M c^{2}}{\\Delta t}=\\frac{10^{5} L_{\\odot}}{M_{\\odot}^{6}} M^{6}.\n\\]\nConverting these to differentials, we get\n\\begin{equation}\n\\frac{0.007 dM c^{2}}{dt}=\\frac{10... |
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... | \left[P_{0}^{2 / 5}-\frac{2}{5} g K^{-3 / 5} z\right]^{5 / 2} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Start with the equation of hydrostatic equilibrium:\n\\[\n\\frac{d P}{d z}=-g \\rho\n\\]\nwhere $g$ is approximately constant through the atmosphere, and is given by $G M / R^{2}$. We can use the polytropic equation of state to eliminate ... |
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. (Consid... | 1.75 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{gathered}\n\\mathcal{F}_{1 \\& 2}=4 \\pi \\sigma\\left(T_{1}^{4} R_{1}^{2}+T_{2}^{4} R_{2}^{2}\\right) \\\\\n\\mathcal{F}_{\\text {eclipse }}=4 \\pi \\sigma T_{1}^{4} R_{1}^{2} \\\\\n\\Delta m=-2.5 \\log \\left(\\frac{\\mathc... |
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 cente... | \sqrt{4 \pi G \rho_{0} r_{0}^{2}} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{gathered}\n-\\frac{G M(<r)}{r^{2}}=-\\frac{v^{2}}{r} \\quad(\\text { from } F=m a) \\\\\nM(<r)=\\int_{0}^{r} \\rho_{0}\\left(\\frac{r_{0}}{r}\\right)^{2} 4 \\pi r^{2} d r=4 \\pi \\rho_{0} r_{0}^{2} r\n\\end{gathered}\n\\]\nNo... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Using the formula for angular resolution $\\theta$ in terms of the effective size $d$ and the wavelength $\\lambda$, namely $\\theta = \\lambda/d$, gives \\boxed{0.33} arcseconds.",
"type": "Introduction to Astronomy (8.282J Spring 2006... |
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 \... | 27.39 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Distance Modulus: $DM=m-M=20.39+7=\\boxed{27.39}$\n\\end{aligned}",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\mathrm{DM}=5 \\log \\left(\\frac{d}{10 \\mathrm{pc}}\\right)=5 \\log (75,000)=\\boxed{24.4}\n\\]",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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... | 0.05 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Using the formula for angular resolution $\\theta$ in terms of the effective size $d$ and the wavelength $\\lambda$, namely $\\theta = \\lambda/d$, gives \\boxed{0.05} arcseconds.",
"type": "Introduction to Astronomy (8.282J Spring 2006... |
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 te... | \arcsin{1.3 \sin{\theta_w}} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Using Snell's law, $1.3 \\sin{\\theta_w} = \\sin{\\theta_a}$. So $\\theta_a = \\boxed{\\arcsin{1.3 \\sin{\\theta_w}}}$.",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\Delta E=\\frac{G M \\Delta m}{R}\n\\]\nThe fractional rest energy lost is $\\Delta E / \\Delta m c^{2}$, or\n\\[\n\\frac{\\Delta E}{\\Delta m c^{2}}=\\frac{G M}{R c^{2}} \\simeq \\boxed{0.15}\n\\]",
"type": "Introduction to Astro... |
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... | 2e27 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{gathered}\n\\ell \\simeq \\frac{1}{n \\sigma}=\\frac{1}{10^{6} \\mathrm{pc}^{-3} \\pi R^{2}} \\\\\n\\ell \\simeq \\frac{1}{3 \\times 10^{-50} \\mathrm{~cm}^{-3} \\times 1.5 \\times 10^{22} \\mathrm{~cm}^{2}} \\simeq \\boxed{2... |
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
... | -1./3 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{gathered}\n\\frac{K \\rho^{5 / 3}}{R} \\simeq\\left(\\frac{G M}{R^{2}}\\right)\\left(\\frac{M}{R^{3}}\\right) \\\\\n\\rho \\sim \\frac{M}{R^{3}} \\\\\n\\frac{K M^{5 / 3}}{R R^{5}} \\simeq \\frac{G M^{2}}{R^{5}} \\\\\nR \\sime... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\lambda \\simeq \\lambda_{0}(1+v / c)\n\\]\nwhere $\\lambda=6784 \\AA$ and $\\lambda_{0}=6565 \\AA$. Rearranging,\n\\[\n\\frac{v}{c} \\simeq \\frac{\\lambda-\\lambda_{0}}{\\lambda_{0}} \\simeq \\frac{6784-6565}{6565} \\Rightarrow v ... |
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 (i... | 0.5613 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The fact that the two sources have the same apparent brightness implies that the flux at the respective distances is the same; since flux varies with distance as $1/d^2$, we find that (with distances in km) $\\frac{3}{3^2} = \\frac{10^{26... |
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... | \frac{v_{0}}{r_{0}} \frac{1}{\left(1+r / r_{0}\right)^{3 / 2}} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$\\omega=v / r & \\Rightarrow \\omega(r)=\\boxed{\\frac{v_{0}}{r_{0}} \\frac{1}{\\left(1+r / r_{0}\\right)^{3 / 2}}}$",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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-... | 2.75e11 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{gathered}\nv_{1}=\\frac{2 \\pi r_{1}}{P_{\\text {orb }}} \\\\\nr_{1}=\\frac{P_{\\text {orb }} v_{1}}{2 \\pi}=\\boxed{2.75e11} \\mathrm{~cm}\n\\end{gathered}\n\\]",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 centimeter... | 6e13 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$\\tau_{\\text {coll }} \\simeq \\frac{2 \\times 10^{27} \\mathrm{~cm}}{10^{6} \\mathrm{~cm} / \\mathrm{sec}} \\simeq 2 \\times 10^{21} \\mathrm{sec} \\simeq \\boxed{6e13} \\text { years }$",
"type": "Introduction to Astronomy (8.282J S... |
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... | 1e-5 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The angular resolution of the full array is given by the wavelength over the dimension of the array, in this case $\\boxed{1e-5}$ radians.",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "At $10 \\mathrm{pc}$, the magnitude is (by definition) just the absolute magnitude of the cluster. Since the total luminosity of the cluster is $10^{6}$ times the luminosity of the Sun, we have that \n\\begin{equation}\n\\delta m = 2.5 \... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Power output goes as $T^4r^2$, so the power output of this star is $\\boxed{15625}$ times that of the Sun.",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\ell=\\frac{1}{n \\sigma}=\\frac{1}{10^{19} 10^{-16}}=\\boxed{1e-3} \\mathrm{~cm}\n\\]",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Total power output goes as $r^2 T^4$, where $r$ is the star's radius, and $T$ is its temperature. Flux, at a distance $R$ away thus goes as $r^2 T^4 / R^2$. In our case, the ratio of flux from star 1 to star 2 is $1/16$ (i.e., star 2 is... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The number of objects detected goes as the cube of the distance for objects with flux greater than a certain minimum flux. At the same time the flux falls off with the inverse square of the distance. Thus, the slope of the $\\log N(>F)$ v... |
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 ... | 2.7778e-6 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The best angular resolution will occur at the maximum separation, and is simply the ratio of wavelength to this separation $p$: $\\theta = \\frac{\\lambda}{p}$, or $\\frac{0.1}{36\\times 10^3}$, which is $\\boxed{2.7778e-6}$ radians.",
... |
A globular cluster has $10^{6}$ stars each of apparent magnitude $+8$. What is the combined apparent magnitude of the entire cluster? | -7 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "\\[\n\\begin{gathered}\n+8=-2.5 \\log \\left(F / F_{0}\\right) \\\\\nF=6.3 \\times 10^{-4} F_{0} \\\\\nF_{\\text {cluster }}=10^{6} \\times 6.3 \\times 10^{-4} F_{0}=630 F_{0} \\\\\nm_{\\text {cluster }}=-2.5 \\log (630)=\\boxed{-7}\n\\en... |
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: Fin... | 48 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Using the Wien displacement law:\n\\[\n \\lambda_{\\max }=0.29 / T \\mathrm{~cm}=\\boxed{48} \\AA\n\\]",
"type": "Introduction to Astronomy (8.282J Spring 2006)"
} |
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 universa... | 16 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "This particular definition of universality only treats arbitrary functions of two Boolean variables, but with any number of outputs. It appears to be an onerous task to prove universality for an arbitrary number of outputs. However, since... |
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 ... | 0.192 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "We see that the probability that a person has the disease given that the test is positive, is:\n\\[\n\\frac{0.001 \\times 0.95}{0.001 \\times 0.95+0.999 \\times 0.004}=19.2 \\%\n\\]\n$\\begin{array}{ccccc}\\text { Have Disease? } & \\text... |
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 ... | 0.5978 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "This is a noisy channel with the same probabilities for mixing up $Z$ and $B$. Channel capacity is defined as the maximum mutual information (for any possible input probability) times the rate $W$. The rate of error is $\\epsilon=0.08$. S... |
Preamble: Given the following data from an Experimental Forest, answer the following questions. Show your work and units.
$\begin{array}{ll}\text { Total vegetative biomass } & 80,000 \mathrm{kcal} \mathrm{m}^{-2} \\ \text { Detritus and organic matter in soil } & 120,000 \mathrm{kcal } \mathrm{m}^{-2} \\ \text { Total... | 11000 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$\\mathrm{NCP}=\\mathrm{GPP}-\\mathrm{R}_{\\mathrm{A}}-\\mathrm{R}_{\\mathrm{H}}=20,000-9000=\\boxed{11000} \\mathrm{kcal} \\mathrm{m}^{-2} \\mathrm{yr}^{-1}$",
"type": "Ecology I (1.018J Fall 2009)"
} |
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 fe... | 0.53 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$N_o = 100$ (in 1990)\n\\\\\n$t = 10$ yr\n\\\\\n$r = 1.3 \\text{yr}^{-1}$\n\\\\\n$t_d = (ln(2))/r = 0.693/(1.3 \\text{yr}^{-1}) = 0.53$ years\n\\\\\nThe doubling time of the ferret population is \\boxed{0.53} years.",
"type": "Ecology I... |
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... | 15000 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "NPP $=$ GPP $-R_{A}=20,000-5,000=\\boxed{15000} \\mathrm{kcal} \\mathrm{m}^{-2} \\mathrm{yr}^{-1}$",
"type": "Ecology I (1.018J Fall 2009)"
} |
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.
Gi... | 1538 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$20,000,000$ tonnes $C / 13,000$ tonnes $C y^{-1}=\\boxed{1538}$ years",
"type": "Ecology I (1.018J Fall 2009)"
} |
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 the... | 4.4e7 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$N_o = 100$ (in 1990)\n\\\\\n$N = ?$ (in 2000)\n\\\\\n$t = 10$ yr\n\\\\\n$r = 1.3 \\text{yr}^{-1}$\n\\\\\n$N = N_{o}e^{rt} = 100*e^{(1.3/\\text{yr})(10 \\text{yr})} = 4.4 x 10^7$ ferrets\n\\\\\nThere will be \\boxed{4.4e7} ferrets on the ... |
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 capacita... | I(0) | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$c=\\boxed{I(0)}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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 r... | y^{\prime}+r y-r x(t)=0 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The differential equation for $y(t)$ is $\\boxed{y^{\\prime}+r y-r x(t)=0}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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 pro... | c x^{-2}-\frac{\cos (2 x)}{2 x^{2}} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "We see that $\\left(x^{2} y\\right)^{\\prime}=x^{2} y^{\\prime}+2 x y$. Thus, $x^{2} y=-\\frac{1}{2} \\cos (2 x)+c$, and $y=\\boxed{c x^{-2}-\\frac{\\cos (2 x)}{2 x^{2}}}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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 ... | \frac{d x}{d t}=k x-a | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The natural growth rate is $k$, meaning that after some short time $\\Delta t$ year(s) passes, we expect $k x(t) \\Delta t$ new oryxes to appear. However, meanwhile the population is reduced by $a \\Delta t$ oryxes due to the harvesting. ... |
If the complex number $z$ is given by $z = 1+\sqrt{3} i$, what is the magnitude of $z^2$? | 4 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$z^{2}$ has argument $2 \\pi / 3$ and radius 4, so by Euler's formula, $z^{2}=4 e^{i 2 \\pi / 3}$. Thus $A=4, \\theta=\\frac{2\\pi}{3}$, so our answer is $\\boxed{4}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
In the polar representation $(r, \theta)$ of the complex number $z=1+\sqrt{3} i$, what is $r$? | 2 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "For z, $r=2$ and $\\theta=\\pi / 3$, so its polar coordinates are $\\left(2, \\frac{\\pi}{3}\\right)$. So $r=\\boxed{2}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Using Euler's formula, we find that the answer is $\\boxed{1+\\sqrt{3} i}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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 ... | 0 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The straight line solution occurs when $c=\\boxed{0}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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}... | -0.875 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$y_3 = \\boxed{-0.875}$",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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) | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Here, our right triangle has hypotenuse $\\sqrt{2}$, so $A=\\sqrt{2}$. Both summands have \"circular frequency\" 2, so $\\omega=2 . \\phi$ is the argument of the hypotenuse, which is $\\pi / 4$, so $f(t)=\\boxed{\\sqrt{2} \\cos (2 t-\\pi ... |
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}) | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "First, notice that both $x(t)=e^{a t}$ and $x(t)=e^{-a t}$ are solutions to $\\ddot{x}-a^{2} x=0$. Then for any constants $c_{1}$ and $c_{2}$, $x(t)=c_{1} e^{a t}+c_{2} e^{-a t}$ are also solutions to $\\ddot{x}-a^{2} x=0$. Moreover, $x(0... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Suppose \\[x(t)=a \\cos (\\omega t)+b \\sin (\\omega t)\\] $x(0)=a$, therefore $a=x_{0}$. Then \\[x^{\\prime}(0)=-a \\omega \\sin 0+b \\omega \\cos 0=b \\omega=\\dot{x}_{0}\\] Then $b=\\dot{x}_{0} / \\omega$. The solution is then $x=\\box... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$1+\\sqrt{3} i$ has modulus 2 and argument $\\pi / 3+2 k \\pi$ for all integers k, so $1+\\sqrt{3} i$ can be expressed as a complex exponential of the form $2 e^{i(\\pi / 3+2 k \\pi)}$. Taking logs gives us the equation $a+b i=\\ln 2+i(\\... |
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^{... | e^{t} / 3 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "When $c=0, x=\\boxed{e^{t} / 3}$ is the solution of the required form.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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... | 2 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Resonance occurs when $\\omega=\\boxed{2}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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 ... | \frac{\cos (2 t)}{15}+C_{1} e^{t}+C_{2} e^{-t}+C_{3} \cos (t)+C_{4} \sin (t) | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "To get the general solution, we take the sum of the general solution to the homogeneous equation and the particular solution to the original equation. The homogeneous equation is $\\frac{d^{4} x}{d t^{4}}-x=0$. The characteristic polynomi... |
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}} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_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 $A=\\boxed{\\frac{1}{4-\\omega^{2}}}$.",
"type": "Diff... |
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} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$\\cos (2 t)=\\operatorname{Re}\\left(e^{2 i t}\\right)$, so $x$ can be the real part of any solution $z$ to $\\dot{z}+2 z=e^{2 i t}$. One solution is given by $x=\\operatorname{Re}\\left(e^{2 i t} /(2+2 i)\\right)=\\boxed{\\frac{\\cos (2... |
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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "We can find this by brute force. If $x=A \\sin (3 t)$, then $\\ddot{x}=-9 A \\sin (3 t)$, so $\\ddot{x}+4 x=-5 A \\sin (3 t)$. Therefore, when $A=\\boxed{-0.2}, x_{p}(t)=-\\sin (3 t) / 5$ is a solution of the given equation.",
"type": "... |
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_... | x / 2-1 / 4+c e^{-2 x} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_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... |
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 $\frac{e^{-2 t}}{p(-2)}=\boxed{\frac{e^{-2 t}}{15}}$... | \frac{e^{-2 t}}{15}+C_{1} e^{t}+C_{2} e^{-t}+ C_{3} \cos (t)+C_{4} \sin (t) | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "To get the general solution, we take the sum of the general solution to the homogeneous equation and the particular solution to the original equation. The homogeneous equation is $\\frac{d^{4} x}{d t^{4}}-x=0$. The characteristic polynomi... |
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}+\omeg... | s^{2}+\omega^{2} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The characteristic polynomial $p(s)$ is $p(s)=\\boxed{s^{2}+\\omega^{2}}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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) | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The right triangle has hypotenuse of length $\\sqrt{1^{2}+(-\\sqrt{3})^{2}}=2$. The circular frequency of both summands is $\\pi$, so $\\omega=\\pi$. The argument of the hypotenuse is $-\\pi / 3$, so $f(t)=\\boxed{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 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The extrema of $x(t)=A e^{-a t} \\cos (\\omega t)$ occur when $\\dot{x}(t)=0$, i.e., $-a \\cos (\\omega t)=\\omega \\sin (\\omega t)$. When $\\omega \\neq 0$, the extrema are achieved at $t$ where $\\tan (\\omega t)=-a / \\omega$. Since m... |
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 ... | \frac{4 A}{3+3 i} e^{i t} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The equation is $\\ddot{x}+3 \\dot{x}+4 x=4 A \\cos t$, with the characteristic polynomial $p(s)=s^{2}+3 s+4$. The complex exponential corresponding to the input signal is $y_{c x}=A e^{i t}$ and $p(i)=3+3 i \\neq 0$. By the Exponential R... |
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 capacita... | I(0) e^{-\frac{t}{R C}}
| {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "When $V$ is constant, the equation becomes $R \\dot{I}+\\frac{1}{C} I=0$, which is separable. Solving gives us\n\\[\nI(t)=\\boxed{I(0) e^{-\\frac{t}{R C}}\n}\\].",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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^{... | \frac{e^{2 i t}}{(2+2 i)} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "When $C=0, z=\\boxed{\\frac{e^{2 i t}}{(2+2 i)}}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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_... | \frac{2-\omega^{2}-\omega i / 4}{\omega^{4}-\frac{63}{16} \omega^{2}+4} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Set $F_{\\mathrm{cx}}=e^{i \\omega t}$. The complex replacement of the equation is $\\ddot{z}+\\frac{1}{4} \\dot{z}+2 z=e^{i \\omega t}$, with the characteristic polynomial $p(s)=s^{2}+\\frac{1}{4} s+2.$ Given that $p(i \\omega)=-\\omega^... |
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 so... | x^{\prime}+\frac{r}{V} x-r c=0 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The concentration of salt at any given time is $x(t) / V \\mathrm{gm} /$ liter, so for small $\\Delta t$, we lose $r x(t) \\Delta t / V$ gm from the exit pipe, and we gain $r c \\Delta t \\mathrm{gm}$ from the input pipe. The equation is ... |
Find the polynomial solution of $\ddot{x}-x=t^{2}+t+1$, solving for $x(t)$. | -t^2 - t - 3 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "Since the constant term of the right-hand side is nonzero, the undetermined coefficients theorem asserts that there is a unique quadratic polynomial $a t^{2}+b t+c$ satisfying this equation. Substituting this form into the left side of th... |
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... | -8 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$e^{n(a+b i)}=(1+\\sqrt{3} i)^{n}$, so the answer is $\\boxed{-8}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
Find a purely sinusoidal solution of $\frac{d^{4} x}{d t^{4}}-x=\cos (2 t)$. | \frac{\cos (2 t)}{15} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_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 solutio... |
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... | -2+2 \sqrt{3} i | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$e^{n(a+b i)}=(1+\\sqrt{3} i)^{n}$, so the answer is $\\boxed{-2+2 \\sqrt{3} i}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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... | \frac{t}{4} \sin (2 t) | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "The complex replacement of the equation is $\\ddot{z}+4 z=e^{2 i t}$, with the characteristic polynomial $p(s)=s^{2}+4$. Because $p(2 i)=0$ and $p^{\\prime}(2 i)=4 i \\neq 0$, we need to use the Resonant ERF, which leads to $z_{p}=\\frac{... |
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}) | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "First, notice that both $x(t)=e^{a t}$ and $x(t)=e^{-a t}$ are solutions to $\\ddot{x}-a^{2} x=0$. Then for any constants $c_{1}$ and $c_{2}$, $x(t)=c_{1} e^{a t}+c_{2} e^{-a t}$ are also solutions to $\\ddot{x}-a^{2} x=0$. Moreover, $x(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. | \frac{e^{t}} {3}+c e^{-2 t} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_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 $x=\\boxed{\\frac{e^{t}} {3}+c e^{-2 t}}$.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
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} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$\\dot{x}=\\dot{u} e^{-t}-u e^{-t}$ and $\\ddot{x}=\\ddot{u} e^{-t}-2 \\dot{u} e^{-t}+u e^{-t}$. Plugging into the equation leads to $e^{-t}(\\ddot{u}+\\dot{u})=t e^{-t}$. Cancelling off $e^{-t}$ from both sides, we get $\\ddot{u}+\\dot{u... |
If the complex number $z$ is given by $z = 1+\sqrt{3} i$, what is the real part of $z^2$? | -2 | {
"dataset": "minerva_math",
"difficulty": "college",
"original_solution": "$z^{2}$ has argument $2 \\pi / 3$ and radius 4 , so by Euler's formula, $z^{2}=4 e^{i 2 \\pi / 3}=-2+2 \\sqrt{3} i$. Thus $a = -2, b = 2\\sqrt 3$, so our answer is \\boxed{-2}.",
"type": "Differential Equations (18.03 Spring 2010)"
} |
Find a purely exponential solution of $\frac{d^{4} x}{d t^{4}}-x=e^{-2 t}$. | \frac{e^{-2 t}}{15} | {
"dataset": "minerva_math",
"difficulty": "college",
"original_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 $\\frac{e^{-2 t}}{p(-2)}=\\boxed{\\frac{e^{-2 t}}{15}}$.",
"type": "D... |
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