Sihangli/scibench-rl huggingface dataset:
Problem: Calculate the change in Gibbs energy of $35 \mathrm{~g}$ of ethanol (mass density $0.789 \mathrm{~g} \mathrm{~cm}^{-3}$ ) when the pressure is increased isothermally from 1 atm to 3000 atm.
Answer:  12  $\mathrm{~kJ}$
Numeric value: 12
Problem: The promotion of an electron from the valence band into the conduction band in pure $\mathrm{TIO}_2$ by light absorption requires a wavelength of less than $350 \mathrm{~nm}$. Calculate the energy gap in electronvolts between the valence and conduction bands.
Answer:  3.54  $\mathrm{eV}$
Numeric value: 3.54
Problem: Although the crystallization of large biological molecules may not be as readily accomplished as that of small molecules, their crystal lattices are no different. Tobacco seed globulin forms face-centred cubic crystals with unit cell dimension of $12.3 \mathrm{~nm}$ and a density of $1.287 \mathrm{~g} \mathrm{~cm}^{-3}$. Determine its molar mass.
Answer: 3.61  $10^5 \mathrm{~g} \mathrm{~mol}^{-1}$
Numeric value: 3.61
Problem: An electron is accelerated in an electron microscope from rest through a potential difference $\Delta \phi=100 \mathrm{kV}$ and acquires an energy of $e \Delta \phi$. What is its final speed?
Answer:  1.88  $10^8 \mathrm{~m} \mathrm{~s}^{-1}$
Numeric value: 1.88
Problem: The following data show how the standard molar constant-pressure heat capacity of sulfur dioxide varies with temperature. By how much does the standard molar enthalpy of $\mathrm{SO}_2(\mathrm{~g})$ increase when the temperature is raised from $298.15 \mathrm{~K}$ to $1500 \mathrm{~K}$ ?
Answer:  62.2  $\mathrm{~kJ} \mathrm{~mol}^{-1}$
Numeric value: 62.2
Problem: Suppose that the normalized wavefunction for an electron in a carbon nanotube of length $L=10.0 \mathrm{~nm}$ is: $\psi=(2 / L)^{1 / 2} \sin (\pi x / L)$. Calculate the probability that the electron is between $x=4.95 \mathrm{~nm}$ and $5.05 \mathrm{~nm}$.
Answer:  0.020  
Numeric value: 0.020
Problem: A sample of the sugar D-ribose of mass $0.727 \mathrm{~g}$ was placed in a calorimeter and then ignited in the presence of excess oxygen. The temperature rose by $0.910 \mathrm{~K}$. In a separate experiment in the same calorimeter, the combustion of $0.825 \mathrm{~g}$ of benzoic acid, for which the internal energy of combustion is $-3251 \mathrm{~kJ} \mathrm{~mol}^{-1}$, gave a temperature rise of $1.940 \mathrm{~K}$. Calculate the enthalpy of formation of D-ribose.
Answer:  -1270  $\mathrm{~kJ} \mathrm{~mol}^{-1}$
Numeric value: -1270
Problem: An electron confined to a metallic nanoparticle is modelled as a particle in a one-dimensional box of length $L$. If the electron is in the state $n=1$, calculate the probability of finding it in the following regions: $0 \leq x \leq \frac{1}{2} L$.
Answer:  $\frac{1}{2}$  
Numeric value: 0.5
Problem: The carbon-carbon bond length in diamond is $154.45 \mathrm{pm}$. If diamond were considered to be a close-packed structure of hard spheres with radii equal to half the bond length, what would be its expected density? The diamond lattice is face-centred cubic and its actual density is $3.516 \mathrm{~g} \mathrm{~cm}^{-3}$.
Answer:  7.654  $\mathrm{~g} \mathrm{~cm}^{-3}$
Numeric value: 7.654
Problem: A swimmer enters a gloomier world (in one sense) on diving to greater depths. Given that the mean molar absorption coefficient of seawater in the visible region is $6.2 \times 10^{-3} \mathrm{dm}^3 \mathrm{~mol}^{-1} \mathrm{~cm}^{-1}$, calculate the depth at which a diver will experience half the surface intensity of light.
Answer:  0.87  $\mathrm{~m}$
Numeric value: 0.87
Problem: Calculate the molar energy required to reverse the direction of an $\mathrm{H}_2 \mathrm{O}$ molecule located $100 \mathrm{pm}$ from a $\mathrm{Li}^{+}$ ion. Take the magnitude of the dipole moment of water as $1.85 \mathrm{D}$.
Answer:  1.07  $10^3 \mathrm{~kJ} \mathrm{~mol}^{-1}$
Numeric value: 1.07
Problem: In an industrial process, nitrogen is heated to $500 \mathrm{~K}$ at a constant volume of $1.000 \mathrm{~m}^3$. The gas enters the container at $300 \mathrm{~K}$ and $100 \mathrm{~atm}$. The mass of the gas is $92.4 \mathrm{~kg}$. Use the van der Waals equation to determine the approximate pressure of the gas at its working temperature of $500 \mathrm{~K}$. For nitrogen, $a=1.39 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}, b=0.0391 \mathrm{dm}^3 \mathrm{~mol}^{-1}$.
Answer:  140  $\mathrm{~atm}$
Numeric value: 140
Problem: The chemical shift of the $\mathrm{CH}_3$ protons in acetaldehyde (ethanal) is $\delta=2.20$ and that of the $\mathrm{CHO}$ proton is 9.80 . What is the difference in local magnetic field between the two regions of the molecule when the applied field is $1.5 \mathrm{~T}$
Answer:  11  $\mu \mathrm{T}$
Numeric value: 11
Problem: Suppose that the junction between two semiconductors can be represented by a barrier of height $2.0 \mathrm{eV}$ and length $100 \mathrm{pm}$. Calculate the transmission probability of an electron with energy $1.5 \mathrm{eV}$.
Answer:  0.8  
Numeric value: 0.8
Problem: The diffusion coefficient of a particular kind of t-RNA molecule is $D=1.0 \times 10^{-11} \mathrm{~m}^2 \mathrm{~s}^{-1}$ in the medium of a cell interior. How long does it take molecules produced in the cell nucleus to reach the walls of the cell at a distance $1.0 \mu \mathrm{m}$, corresponding to the radius of the cell?
Answer:  1.7  $10^{-2} \mathrm{~s}$
Numeric value: 1.7
Problem: At what pressure does the mean free path of argon at $20^{\circ} \mathrm{C}$ become comparable to the diameter of a $100 \mathrm{~cm}^3$ vessel that contains it? Take $\sigma=0.36 \mathrm{~nm}^2$
Answer:  0.195  $\mathrm{Pa}$
Numeric value: 0.195
Problem: The equilibrium pressure of $\mathrm{O}_2$ over solid silver and silver oxide, $\mathrm{Ag}_2 \mathrm{O}$, at $298 \mathrm{~K}$ is $11.85 \mathrm{~Pa}$. Calculate the standard Gibbs energy of formation of $\mathrm{Ag}_2 \mathrm{O}(\mathrm{s})$ at $298 \mathrm{~K}$.
Answer:  -11.2  $\mathrm{~kJ}   \mathrm{~mol}^{-1}$
Numeric value: -11.2
Problem: When alkali metals dissolve in liquid ammonia, their atoms each lose an electron and give rise to a deep-blue solution that contains unpaired electrons occupying cavities in the solvent. These 'metal-ammonia solutions' have a maximum absorption at $1500 \mathrm{~nm}$. Supposing that the absorption is due to the excitation of an electron in a spherical square well from its ground state to the next-higher state (see the preceding problem for information), what is the radius of the cavity?
Answer:  0.69  $\mathrm{~nm}$
Numeric value: 0.69
Problem: Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \mathrm{pm}$ ? 
Answer:  7.27  $10^6 \mathrm{~m} \mathrm{~s}^{-1}$
Numeric value: 7.27
Problem: Nelson, et al. (Science 238, 1670 (1987)) examined several weakly bound gas-phase complexes of ammonia in search of examples in which the $\mathrm{H}$ atoms in $\mathrm{NH}_3$ formed hydrogen bonds, but found none. For example, they found that the complex of $\mathrm{NH}_3$ and $\mathrm{CO}_2$ has the carbon atom nearest the nitrogen (299 pm away): the $\mathrm{CO}_2$ molecule is at right angles to the $\mathrm{C}-\mathrm{N}$ 'bond', and the $\mathrm{H}$ atoms of $\mathrm{NH}_3$ are pointing away from the $\mathrm{CO}_2$. The magnitude of the permanent dipole moment of this complex is reported as $1.77 \mathrm{D}$. If the $\mathrm{N}$ and $\mathrm{C}$ atoms are the centres of the negative and positive charge distributions, respectively, what is the magnitude of those partial charges (as multiples of $e$ )?
Answer:  0.123  
Numeric value: 0.123
Problem: The NOF molecule is an asymmetric rotor with rotational constants $3.1752 \mathrm{~cm}^{-1}, 0.3951 \mathrm{~cm}^{-1}$, and $0.3505 \mathrm{~cm}^{-1}$. Calculate the rotational partition function of the molecule at $25^{\circ} \mathrm{C}$.
Answer:  7.97 $10^3$
Numeric value: 7.97
Problem: Suppose that $2.5 \mathrm{mmol} \mathrm{N}_2$ (g) occupies $42 \mathrm{~cm}^3$ at $300 \mathrm{~K}$ and expands isothermally to $600 \mathrm{~cm}^3$. Calculate $\Delta G$ for the process.
Answer:  -17  $\mathrm{~J}$ 
Numeric value: -17
Problem: Calculate the standard potential of the $\mathrm{Ce}^{4+} / \mathrm{Ce}$ couple from the values for the $\mathrm{Ce}^{3+} / \mathrm{Ce}$ and $\mathrm{Ce}^{4+} / \mathrm{Ce}^{3+}$ couples.

Answer:  -1.46  $\mathrm{V}$
Numeric value: -1.46
Problem: An effusion cell has a circular hole of diameter $1.50 \mathrm{~mm}$. If the molar mass of the solid in the cell is $300 \mathrm{~g} \mathrm{~mol}^{-1}$ and its vapour pressure is $0.735 \mathrm{~Pa}$ at $500 \mathrm{~K}$, by how much will the mass of the solid decrease in a period of $1.00 \mathrm{~h}$ ?
Answer:  16  $\mathrm{mg}$
Numeric value: 16
Problem: The speed of a certain proton is $6.1 \times 10^6 \mathrm{~m} \mathrm{~s}^{-1}$. If the uncertainty in its momentum is to be reduced to 0.0100 per cent, what uncertainty in its location must be tolerated?
Answer:  52  $\mathrm{pm}$
Numeric value: 52
Problem: It is possible to produce very high magnetic fields over small volumes by special techniques. What would be the resonance frequency of an electron spin in an organic radical in a field of $1.0 \mathrm{kT}$ ?
Answer:  2.8  $10^{13} \mathrm{~Hz}$
Numeric value: 2.8
Problem: A particle of mass $1.0 \mathrm{~g}$ is released near the surface of the Earth, where the acceleration of free fall is $g=8.91 \mathrm{~m} \mathrm{~s}^{-2}$. What will be its kinetic energy after  $1.0 \mathrm{~s}$. Ignore air resistance?
Answer: 48  $\mathrm{~mJ}$
Numeric value: 48
Problem: The flux of visible photons reaching Earth from the North Star is about $4 \times 10^3 \mathrm{~mm}^{-2} \mathrm{~s}^{-1}$. Of these photons, 30 per cent are absorbed or scattered by the atmosphere and 25 per cent of the surviving photons are scattered by the surface of the cornea of the eye. A further 9 per cent are absorbed inside the cornea. The area of the pupil at night is about $40 \mathrm{~mm}^2$ and the response time of the eye is about $0.1 \mathrm{~s}$. Of the photons passing through the pupil, about 43 per cent are absorbed in the ocular medium. How many photons from the North Star are focused onto the retina in $0.1 \mathrm{~s}$ ?
Answer:  4.4  $10^3$
Numeric value: 4.4
Problem: When ultraviolet radiation of wavelength $58.4 \mathrm{~nm}$ from a helium lamp is directed on to a sample of krypton, electrons are ejected with a speed of $1.59 \times 10^6 \mathrm{~m} \mathrm{~s}^{-1}$. Calculate the ionization energy of krypton.

Answer:  14  $\mathrm{eV}$
Numeric value: 14
Problem:  If $125 \mathrm{~cm}^3$ of hydrogen gas effuses through a small hole in 135 seconds, how long will it take the same volume of oxygen gas to effuse under the same temperature and pressure?
Answer:  537  $\mathrm{s}$
Numeric value: 537
Problem: The vibrational wavenumber of $\mathrm{Br}_2$ is $323.2 \mathrm{~cm}^{-1}$. By evaluating the vibrational partition function explicitly (without approximation), at what temperature is the value within 5 per cent of the value calculated from the approximate formula?
Answer:  4500 $\mathrm{~K}$
Numeric value: 4500
Problem: A thermodynamic study of $\mathrm{DyCl}_3$ (E.H.P. Cordfunke, et al., J. Chem. Thermodynamics 28, 1387 (1996)) determined its standard enthalpy of formation from the following information
(1) $\mathrm{DyCl}_3(\mathrm{~s}) \rightarrow \mathrm{DyCl}_3(\mathrm{aq}$, in $4.0 \mathrm{M} \mathrm{HCl}) \quad \Delta_{\mathrm{r}} H^{\ominus}=-180.06 \mathrm{~kJ} \mathrm{~mol}^{-1}$
(2) $\mathrm{Dy}(\mathrm{s})+3 \mathrm{HCl}(\mathrm{aq}, 4.0 \mathrm{~m}) \rightarrow \mathrm{DyCl}_3(\mathrm{aq}$, in $4.0 \mathrm{M} \mathrm{HCl}(\mathrm{aq}))+\frac{3}{2} \mathrm{H}_2(\mathrm{~g})$ $\Delta_{\mathrm{r}} H^{\ominus}=-699.43 \mathrm{~kJ} \mathrm{~mol}^{-1}$
(3) $\frac{1}{2} \mathrm{H}_2(\mathrm{~g})+\frac{1}{2} \mathrm{Cl}_2(\mathrm{~g}) \rightarrow \mathrm{HCl}(\mathrm{aq}, 4.0 \mathrm{M}) \quad \Delta_{\mathrm{r}} H^{\ominus}=-158.31 \mathrm{~kJ} \mathrm{~mol}^{-1}$
Determine $\Delta_{\mathrm{f}} H^{\ominus}\left(\mathrm{DyCl}_3, \mathrm{~s}\right)$ from these data.
Answer:  -994.3  $\mathrm{~kJ} \mathrm{~mol}^{-1}$
Numeric value: -994.3
Problem: Calculate $\Delta_{\mathrm{r}} G^{\ominus}(375 \mathrm{~K})$ for the reaction $2 \mathrm{CO}(\mathrm{g})+\mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_2(\mathrm{~g})$ from the values of $\Delta_{\mathrm{r}} G^{\ominus}(298 \mathrm{~K})$ : and $\Delta_{\mathrm{r}} H^{\ominus}(298 \mathrm{~K})$, and the GibbsHelmholtz equation.
Answer: -501 $\mathrm{~kJ} \mathrm{~mol}^{-1}$
Numeric value: -501
Problem: The vapour pressure of benzene is $53.3 \mathrm{kPa}$ at $60.6^{\circ} \mathrm{C}$, but it fell to $51.5 \mathrm{kPa}$ when $19.0 \mathrm{~g}$ of an non-volatile organic compound was dissolved in $500 \mathrm{~g}$ of benzene. Calculate the molar mass of the compound.
Answer:  85  $\mathrm{~g} \mathrm{~mol}^{-1}$
Numeric value: 85
Problem: J.G. Dojahn, et al. (J. Phys. Chem. 100, 9649 (1996)) characterized the potential energy curves of the ground and electronic states of homonuclear diatomic halogen anions. The ground state of $\mathrm{F}_2^{-}$ is ${ }^2 \sum_{\mathrm{u}}^{+}$ with a fundamental vibrational wavenumber of $450.0 \mathrm{~cm}^{-1}$ and equilibrium internuclear distance of $190.0 \mathrm{pm}$. The first two excited states are at 1.609 and $1.702 \mathrm{eV}$ above the ground state. Compute the standard molar entropy of $\mathrm{F}_2^{-}$ at $ 298 \mathrm{~K}$.
Answer:  199.4  $\mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$
Numeric value: 199.4
Problem: The duration of a $90^{\circ}$ or $180^{\circ}$ pulse depends on the strength of the $\mathscr{B}_1$ field. If a $180^{\circ}$ pulse requires $12.5 \mu \mathrm{s}$, what is the strength of the $\mathscr{B}_1$ field? 
Answer:  5.9  $10^{-4} \mathrm{~T}$
Numeric value: 5.9
Problem: In 1976 it was mistakenly believed that the first of the 'superheavy' elements had been discovered in a sample of mica. Its atomic number  was believed to be 126. What is the most probable distance of the innermost electrons from the nucleus of an atom of this element? (In such elements, relativistic effects are very important, but ignore them here.)
Answer:  0.42  $\mathrm{pm}$
Numeric value: 0.42
Problem: The ground level of $\mathrm{Cl}$ is ${ }^2 \mathrm{P}_{3 / 2}$ and a ${ }^2 \mathrm{P}_{1 / 2}$ level lies $881 \mathrm{~cm}^{-1}$ above it. Calculate the electronic contribution to the molar Gibbs energy of $\mathrm{Cl}$ atoms at  $500 \mathrm{~K}$.
Answer:  -6.42  $\mathrm{~kJ} \mathrm{~mol}^{-1}$
Numeric value: -6.42
Problem: Calculate the melting point of ice under a pressure of 50 bar. Assume that the density of ice under these conditions is approximately $0.92 \mathrm{~g} \mathrm{~cm}^{-3}$ and that of liquid water is $1.00 \mathrm{~g} \mathrm{~cm}^{-3}$.
Answer:  272.8  $\mathrm{K}$
Numeric value: 272.8
Problem: What is the temperature of a two-level system of energy separation equivalent to $400 \mathrm{~cm}^{-1}$ when the population of the upper state is one-third that of the lower state?
Answer:  524  $ \mathrm{~K}$
Numeric value: 524
Problem: At $300 \mathrm{~K}$ and $20 \mathrm{~atm}$, the compression factor of a gas is 0.86 . Calculate the volume occupied by $8.2 \mathrm{mmol}$ of the gas under these conditions.
Answer:  8.7  $\mathrm{~cm}^3$
Numeric value: 8.7
Problem: A very crude model of the buckminsterfullerene molecule $\left(\mathrm{C}_{60}\right)$ is to treat it as a collection of electrons in a cube with sides of length equal to the mean diameter of the molecule $(0.7 \mathrm{~nm})$. Suppose that only the $\pi$ electrons of the carbon atoms contribute, and predict the wavelength of the first excitation of $\mathrm{C}_{60}$. (The actual value is $730 \mathrm{~nm}$.)
Answer:  1.6  $\mu \mathrm{m}$
Numeric value: 1.6
Problem: Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.
Answer: 7.25  %
Numeric value: 7.25
Problem: Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains $200 \mathrm{~L}$ of a dye solution with a concentration of $1 \mathrm{~g} / \mathrm{L}$. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of $2 \mathrm{~L} / \mathrm{min}$, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches $1 \%$ of its original value.
Answer:  460.5  min
Numeric value: 460.5
Problem: A certain vibrating system satisfies the equation $u^{\prime \prime}+\gamma u^{\prime}+u=0$. Find the value of the damping coefficient $\gamma$ for which the quasi period of the damped motion is $50 \%$ greater than the period of the corresponding undamped motion.
Answer:  1.4907  
Numeric value: 1.4907
Problem: Find the value of $y_0$ for which the solution of the initial value problem
$$
y^{\prime}-y=1+3 \sin t, \quad y(0)=y_0
$$
remains finite as $t \rightarrow \infty$
Answer:  -2.5  
Numeric value:  -2.5
Problem: A certain spring-mass system satisfies the initial value problem
$$
u^{\prime \prime}+\frac{1}{4} u^{\prime}+u=k g(t), \quad u(0)=0, \quad u^{\prime}(0)=0
$$
where $g(t)=u_{3 / 2}(t)-u_{5 / 2}(t)$ and $k>0$ is a parameter.
Suppose $k=2$. Find the time $\tau$ after which $|u(t)|<0.1$ for all $t>\tau$.
Answer:  25.6773  
Numeric value: 25.6773
Problem: Suppose that a sum $S_0$ is invested at an annual rate of return $r$ compounded continuously.
Determine $T$ if $r=7 \%$.
Answer:  9.90  year
Numeric value: 9.90
Problem: A mass weighing $2 \mathrm{lb}$ stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, by determining the position $u$ of the mass at any time $t$, find the frequency of the motion
Answer:  $\pi/4$  s
Numeric value: 0.7854
Problem: If $\mathbf{x}=\left(\begin{array}{c}2 \\ 3 i \\ 1-i\end{array}\right)$ and $\mathbf{y}=\left(\begin{array}{c}-1+i \\ 2 \\ 3-i\end{array}\right)$, find $(\mathbf{y}, \mathbf{y})$.
Answer:  16  
Numeric value: 16
Problem: Consider the initial value problem
$$
4 y^{\prime \prime}+12 y^{\prime}+9 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-4 .
$$
Determine where the solution has the value zero.
Answer:  0.4  
Numeric value:  0.4
Problem: A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. What monthly payment rate is required to pay off the loan in 3 years?
Answer:  258.14  $
Numeric value:  258.14
Problem: Consider the initial value problem
$$
y^{\prime \prime}+\gamma y^{\prime}+y=k \delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0
$$
where $k$ is the magnitude of an impulse at $t=1$ and $\gamma$ is the damping coefficient (or resistance).
Let $\gamma=\frac{1}{2}$. Find the value of $k$ for which the response has a peak value of 2 ; call this value $k_1$.
Answer:  2.8108  
Numeric value: 2.8108
Problem: If a series circuit has a capacitor of $C=0.8 \times 10^{-6} \mathrm{~F}$ and an inductor of $L=0.2 \mathrm{H}$, find the resistance $R$ so that the circuit is critically damped.
Answer:  1000  $\Omega$
Numeric value: 1000
Problem: If $y_1$ and $y_2$ are a fundamental set of solutions of $t y^{\prime \prime}+2 y^{\prime}+t e^t y=0$ and if $W\left(y_1, y_2\right)(1)=2$, find the value of $W\left(y_1, y_2\right)(5)$.
Answer:  2/25  
Numeric value: 0.08
Problem: Consider the initial value problem
$$
5 u^{\prime \prime}+2 u^{\prime}+7 u=0, \quad u(0)=2, \quad u^{\prime}(0)=1
$$
Find the smallest $T$ such that $|u(t)| \leq 0.1$ for all $t>T$.
Answer: 14.5115  
Numeric value: 14.5115
Problem: Consider the initial value problem
$$
y^{\prime}=t y(4-y) / 3, \quad y(0)=y_0
$$
Suppose that $y_0=0.5$. Find the time $T$ at which the solution first reaches the value 3.98.
Answer:  3.29527  
Numeric value: 3.29527
Problem: Consider the initial value problem
$$
2 y^{\prime \prime}+3 y^{\prime}-2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-\beta,
$$
where $\beta>0$.
Find the smallest value of $\beta$ for which the solution has no minimum point.
Answer:  2  
Numeric value: 2
Problem: Consider the initial value problem
$$
y^{\prime}=t y(4-y) /(1+t), \quad y(0)=y_0>0 .
$$
If $y_0=2$, find the time $T$ at which the solution first reaches the value 3.99.
Answer:  2.84367  
Numeric value: 2.84367
Problem: A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$.
If the mass is to attain a velocity of no more than $10 \mathrm{~m} / \mathrm{s}$, find the maximum height from which it can be dropped.
Answer:  13.45  m
Numeric value:  13.45
Problem: A home buyer can afford to spend no more than $\$ 800$ /month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.
Determine the maximum amount that this buyer can afford to borrow.
Answer:  89,034.79  $
Numeric value: 89,034.79
Problem: A spring is stretched 6 in by a mass that weighs $8 \mathrm{lb}$. The mass is attached to a dashpot mechanism that has a damping constant of $0.25 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}$ and is acted on by an external force of $4 \cos 2 t \mathrm{lb}$.
If the given mass is replaced by a mass $m$, determine the value of $m$ for which the amplitude of the steady state response is maximum.
Answer:  4  slugs
Numeric value: 4
Problem: A recent college graduate borrows $\$ 100,000$ at an interest rate of $9 \%$ to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $800(1+t / 120)$, where $t$ is the number of months since the loan was made.
Assuming that this payment schedule can be maintained, when will the loan be fully paid?
Answer:  135.36  months
Numeric value:  135.36
Problem: Consider the initial value problem
$$
y^{\prime}+\frac{1}{4} y=3+2 \cos 2 t, \quad y(0)=0
$$
Determine the value of $t$ for which the solution first intersects the line $y=12$.
Answer:  10.065778  
Numeric value: 10.065778
Problem: An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of \$25 per month. Find the balance in the account after 3 years.
Answer:  2283.63  $
Numeric value: 2283.63
Problem: A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$.
If the mass is dropped from a height of $30 \mathrm{~m}$, find its velocity when it hits the ground.
Answer:  11.58  m/s
Numeric value:  11.58
Problem: A mass of $100 \mathrm{~g}$ stretches a spring $5 \mathrm{~cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10 \mathrm{~cm} / \mathrm{s}$, and if there is no damping, determine when does the mass first return to its equilibrium position.
Answer:  $\pi/14$  s
Numeric value: 0.2244
Problem: Suppose that a tank containing a certain liquid has an outlet near the bottom. Let $h(t)$ be the height of the liquid surface above the outlet at time $t$. Torricelli's principle states that the outflow velocity $v$ at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height $h$.
Consider a water tank in the form of a right circular cylinder that is $3 \mathrm{~m}$ high above the outlet. The radius of the tank is $1 \mathrm{~m}$ and the radius of the circular outlet is $0.1 \mathrm{~m}$. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet.
Answer:  130.41  s
Numeric value: 130.41
Problem: Solve the initial value problem $y^{\prime \prime}-y^{\prime}-2 y=0, y(0)=\alpha, y^{\prime}(0)=2$. Then find $\alpha$ so that the solution approaches zero as $t \rightarrow \infty$.
Answer:  −2  
Numeric value: −2
Problem: If $y_1$ and $y_2$ are a fundamental set of solutions of $t^2 y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ and if $W\left(y_1, y_2\right)(2)=3$, find the value of $W\left(y_1, y_2\right)(4)$.
Answer:  4.946  
Numeric value: 4.946