role stringclasses 1
value | content stringlengths 80 4.77k | ground_truth stringlengths 0 1.45k |
|---|---|---|
user | Present a modified reaction equation for the combustion of 1 mole of sucrose {eq}(C_{12}H_{22}O_{11})
{/eq} with oxygen gas {eq}(O_2)
{/eq} such that only 10% of the mole of sucrose burns and 6 of every 12 atoms of carbon from that 10% would remain as solid residue after combustion, using decimal fractions for coeffi... | 0.1C_{12}H_{22}O_{11}(s) + 0.6O_2 -> 0.6C(s) + 0.6CO_2(g) + 1.1H_2O
34.2g 19.2g 7.20g 26.4g 19.8g |
user | Peak bone mass typically occurs when a person is in his or her
a. teens
b. twenties
c. thirties
d. fifties Please reason step by step, and put your final answer within \boxed{}. | b |
user | Consider the titration of 100.0 mL of 0.575 M ammonia with 0.500 M HCl. Calculate the pH after 60.0 mL of HCl has been added. Please reason step by step, and put your final answer within \boxed{}. | 9.217 |
user | The motor for a toy Ferris wheel creates a torque of 15 Nm. The wheel requires 29 seconds to reach its design frequency of one revolution per 12 seconds. What is the moment of inertia of the wheel? Please reason step by step, and put your final answer within \boxed{}. | 833.3 kg⋅m^2 |
user | Find the eigenvalues of the following matrix:
$\left(
\begin{array}{cc}
-\frac{29}{3} & -\frac{4}{3} \\
-\frac{2}{3} & 8 \\
\end{array}
\right)$. Please reason step by step, and put your final answer within \boxed{}. | $\lambda = \frac{\sqrt{\frac{947}{3}}}{2}-\frac{5}{6}$ or $\lambda =-\frac{5}{6}-\frac{\sqrt{\frac{947}{3}}}{2}$ |
user | How do I integrate $$ \frac{1}{v}(v^2-c_s^2)\frac{dv}{dr} = 2c_s^2[\frac{r-r_c}{r^2}] $$ to get $$ (\frac{v}{c_s})^2 - \ln(\frac{v}{c_s})^2 = 4\ln(\frac{r}{r_c})+4\frac{r_c}{r}+C $$ given that I've reached $$ \frac{v^2}{c_s^2} - 2\ln(v)=4\frac{r_c}{r}-4\ln(r) $$? Also, do I need to sub in limits and if so, what should ... | v^2/c_s^2 - \ln(v/c_s)^2 = 4\ln(r/r_c) + 4r_c/r + C |
user | A 33.0-gram iron rod, initially at 21.6 degrees Celsius, is submerged into an unknown mass of water at 63.3 degrees Celsius in an insulated container. If the final temperature of the mixture upon reaching thermal equilibrium is 59.7 degrees Celsius, what is the mass of the water? Please reason step by step, and put you... | 37.5 g |
user | An 67.6 kg object moving to the right at 38.7 cm/s overtakes and collides elastically with a second 48.9 kg object moving in the same direction at 19.2 cm/s. Find the velocity of the second object after the collision. Answer in units of cm/s Please reason step by step, and put your final answer within \boxed{}. | 41.7 |
user | Citric acid, the compound responsible for the bitter taste in lemons, has the following composition: C = 37.51%, H = 4.40%, and O = 58.29%. Calculate the empirical formula. Please reason step by step, and put your final answer within \boxed{}. | C6H8O7 |
user | Statistical regression in experiments is the tendency for
(a) research participants who are near the 50th percentile on the pretest to show smaller gains on the posttest than research participants between the 1st and 20th percentiles.
(b) research participants who are at the 95th or higher percentile on the pretest t... | b |
user | Here is the question I am trying to solve letter $(b)$ of it: Here is a solution to it: Which is a very very long solution. Does anyone have a more elegant and succinct solution please? Please reason step by step, and put your final answer within \boxed{}. | β Tr(M) |
user | Find a differential equation for $w(t) = \phi_t(z) = \frac{z+ \tan t}{1 - z(\tan t)}$. Please reason step by step, and put your final answer within \boxed{}. | $w'(z)(1 + z^2) = 1 + (w(z))^2.$ |
user | I am trying to find a solution to $893x - 2432y = 19$ where both $x$ and $y$ are positive integers. When I apply the extended Euclidean algorithm I get a solution where both integers are negative $(-49,-18)$, but in my book the solutions are described as $(79,29)$. Why are my solutions different, given my initial step ... | (79, 29) |
user | From a height of 35.5 m, a 1.30 kg bird dives (from rest) into a small fish tank containing 50.5 kg of water. What is the maximum rise in temperature of the water if the bird gives it all of its mechanical energy? Please reason step by step, and put your final answer within \boxed{}. | 0.002 ^oC |
user | What is the probability that at least 15% of the 110 incoming economics majors change their major before graduation, given that 20% of economics majors at the college change majors before they graduate? Please reason step by step, and put your final answer within \boxed{}. | 0.905 |
user | Construct a 95% confidence interval to determine if there is a difference in the proportion of individuals who believe it is morally wrong for unwed women to have children. The lower bound is_. Please reason step by step, and put your final answer within \boxed{}. | 0.028 |
user | How can the right-hand side of the equation $\frac{p(O_{fg} \mid I, I_t)}{p(O_{bg} \mid I, I_t)} = \frac{p(I \mid I_t, O_{fg}) p(O_{fg})}{p(I \mid I_t, O_{bg}) p(O_{bg})}$ be derived from the left-hand side using Bayes' rule? Please reason step by step, and put your final answer within \boxed{}. | \frac{p(I \mid O_{fg}, I_t)p(O_{fg})}{p(I \mid O_{bg}, I_t) p(O_{bg})} |
user | Solve {eq}\int \frac{dx}{(x + 2) \sqrt{x^2 + 4x + 3}}{/eq} Please reason step by step, and put your final answer within \boxed{}. | \arctan(\sqrt{x^2 + 4x + 3}) + C |
user | What is a solution to the differential equation dy/dx=x/(x+y)? Please reason step by step, and put your final answer within \boxed{}. | #G(y(x),x,C)=0# |
user | How is this equation in the left-hand side, approximately equal to the right hand side? What math technique is used? $$k\left(\sqrt{r_o^2+x^2}-r_o\right) \approx kr_o \left(1+\frac12 \frac{x^2}{r_o^2}-1\right)$$ Please reason step by step, and put your final answer within \boxed{}. | $kr_o \left(1+\frac12 \frac{x^2}{r_o^2}-1\right)$ |
user | If you have 50 cc of a 40 mg/dL solution, how much water do you need to add to dilute the solution to 17 mg/dL? Please reason step by step, and put your final answer within \boxed{}. | 67.55 cc |
user | Which Hardy-Weinberg equation relates the frequencies of the phenotypes for a gene? Please reason step by step, and put your final answer within \boxed{}. | p^2 + 2pq + q^2 = 1 |
user | What is the function of the structure identified by the red arrow?
O Food capture
O Mobility
DNA transfer
O Protection Please reason step by step, and put your final answer within \boxed{}. | D |
user | Solve the initial-value problem.
{eq}x \frac{dy}{dx} - y = 3 x^2, y(9) = 36{/eq} Please reason step by step, and put your final answer within \boxed{}. | y=3x^2-77x |
user | Consider the following congruences system: $$3x \equiv 1 \pmod{8} \\ x \equiv 7 \pmod {12} \\ x \equiv 4 \pmod {15}$$ Find a minimal solution for the system. Please reason step by step, and put your final answer within \boxed{}. | x≡19(mod 120) |
user | How do you find two unit vectors which are orthogonal to the vector 3i-j+2k and parallel to the yz plane? Please reason step by step, and put your final answer within \boxed{}. | \#1/sqrt 13(2, 0, -3) and 1/sqrt 3(1, 1, -1)\# |
user | Use cylindrical coordinates to describe the set of points (x,y,z) with y ≤ 0, x^2+y^2 ≤ 4, and 1 ≤ z ≤ 4 Please reason step by step, and put your final answer within \boxed{}. | {eq}0 \leq r \leq 2, \: \pi \leq \theta \leq 2\pi, \: 1 \leq z \leq 4.{/eq} |
user | How to evaluate the integral $\int_{0}^{\infty}\frac{e^{-x^2}}{x}dx$? Please reason step by step, and put your final answer within \boxed{}. | \frac{1}{2}\gamma |
user | Determine the pH of a 0.28 M methylammonium chloride, {eq}\rm CH_3NH_3Cl
{/eq} solution. Please reason step by step, and put your final answer within \boxed{}. | 5.65 |
user | Suppose we have $f=x^5-1$ and $g=x^2-1$ and I have found the gcd using the Euclidean algorithm but I’m trying to find a way of expressing this in the way $$af+bg=1$$ where $f, g \in Q[x] $. I have the following $x^5-1=(x^2-1)(x^3*x)+(x-1) $ $x^3+x=(x-1)(x^2+x+2)+2 $ So gcd is 1 but how can I work backwards to find how ... | \((x^5-1)(1)-(x^2-1)(x^3+x) = x-1\) |
user | A soccer ball is kicked and given an initial velocity of {eq}12\, m/s
{/eq} at an angle of {eq}30
{/eq} degrees from the ground, what's the velocity of the ball in the x-axis? Please reason step by step, and put your final answer within \boxed{}. | 10 m/s |
user | Determine the Taylor Polynomial of Degree 3, centered at {eq}\displaystyle x=1 {/eq} for {eq}\displaystyle f(x) = 6x+e^{-x} {/eq}. Please reason step by step, and put your final answer within \boxed{}. | T_3(x) = 6.36788+5.632121(x-1)+0.016722(x-1)^2 -0.061313(x-1)^3 |
user | solve $\log x=\frac{1}{2}(x-1)$ What is my mistake in the following steps that only allows me to find one solution (apparently with $W_0$), but I can't find the solution with $W_{-1}$ this way? $$\log x=\frac{1}{2}(x-1)$$ $$\log x-\frac{1}{2}x=-\frac{1}{2}$$ $$e^{\log(x)-\frac{1}{2}x}=e^{-\frac{1}{2}}$$ $$xe^{-\frac{1}... | $x=-2W(-\frac{1}{2}e^{-\frac{1}{2}})$ |
user | Let {eq}\Omega{/eq} be the annulus {eq}1\leq \sqrt{x^2+y^2} \leq 2{/eq}. Let {eq}\partial \Omega{/eq} denote the boundary of the annulus (two concentric circles), and {eq}\vec{n}{/eq} be the normal vector field along {eq}\partial \Omega{/eq} pointing outside. Let {eq}\vec{E}{/eq} be the gradient vector field of the fun... | 0 |
user | The state of a hydrogen atom is described by the following superposition {eq}\displaystyle \Psi = A(1 \psi_{(1,0,0)} - 2 \psi_{2,0,0} -2 \psi_{3,2,2} )
{/eq}
What is the expectation value of {eq}\displaystyle \hat{H}
{/eq}? Please reason step by step, and put your final answer within \boxed{}. | -3.7 eV |
user | Find the maximum and minimum value of the function {eq}\displaystyle f(x,\ y) = x^2 + y^2 - 3 y {/eq} on the closed disk {eq}x^2 + y^2 \le 4 {/eq}. Please reason step by step, and put your final answer within \boxed{}. | -2.25 |
user | A survey claims that {eq}9
{/eq} out of {eq}10
{/eq} doctors recommend aspirin for their patients with headaches. To test this claim against the alternative that the actual proportion of doctors who recommend aspirin is less than {eq}0.90
{/eq}, a random sample of {eq}100
{/eq} doctors results in {eq}83
{/eq} who ... | b |
user | Evaluate the following integral
{eq}\displaystyle \int \frac{\cos^3(x)}{\sqrt\sin (x)} dx {/eq} Please reason step by step, and put your final answer within \boxed{}. | {eq}2\sqrt{\sin x} - \frac{2}{5}\sin x^{\frac{5}{2}} + C{/eq} |
user | 85.65 grams of barium hydroxide was dissolved in 2000 mL of water. 88 grams of carbon dioxide was passed through the solution producing a white precipitate. What was this white precipitate and how many grams of it were produced? Please reason step by step, and put your final answer within \boxed{}. | 98.452 g of barium carbonate |
user | A car is going {eq}\displaystyle
44m/s
{/eq} and brakes and stops in {eq}\displaystyle
3.2
{/eq} seconds, how far did it go before stopping? Please reason step by step, and put your final answer within \boxed{}. | 140.8 meters |
user | Let $p$ be a prime number. How many triples of integers $(x,y,z)$ satisfies the ecuation $$p(xy+xz+yz-p(x+y+z))=xyz?$$ Please reason step by step, and put your final answer within \boxed{}. | 40 |
user | The line segment joining the points (0,5) and (12,12) is revolved about the x-axis to generate a frustum of a cone. Find the surface area of the frustum using the parameterization \displaystyle x=\frac{12}{7} t,\ y = t + 5,\ \ 0 \leq t \leq 7. Please reason step by step, and put your final answer within \boxed{}. | 640.88 |
user | Select the best response: a 95% confidence interval for a mean is 20.91 to 1.36. From this we can infer with 95% confidence that the a. population mean is 0. b. population mean is greater than 0. c. population mean is greater than 20.91. Please reason step by step, and put your final answer within \boxed{}. | b |
user | What is the most likely inheritance pattern of a trait with Gaussian proportional distribution of phenotypic forms? Please reason step by step, and put your final answer within \boxed{}. | polygenic inheritance |
user | Which of the following does not describe the process of summation?
a. Two ESPSs are generated at the same time by two separate synapses, bringing the cell to threshold.
b. One EPSP and one IPSP cancel each other out.
c. Two EPSPs occur close enough in time to bring the cell to threshold.
d. One EPSP fails to bring ... | d |
user | For what values of a is g continuous?
g(x) = {eq}\left\{\begin{matrix}
\frac{1 - cos x}{3x}& if & x \neq 0\\
a & if & x = 0
\end{matrix}\right.
{/eq}
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
O A. a = ---- (Use a comma to separate answers as needed.)
O ... | A |
user | A baseball {eq}(m = 145g)
{/eq} traveling 40 m/s moves a fielder's glove backward 27 cm when the ball is caught. What was the average force exerted by the ball on the glove? Please reason step by step, and put your final answer within \boxed{}. | 429.63 N |
user | Find the point on the plane 2x + 2y + z = 27 that is closest to the origin by minimizing the square of the distance.
x = \boxed{\space}
y = \boxed{\space}
z = \boxed{\space} Please reason step by step, and put your final answer within \boxed{}. | (6, 6, 3) |
user | Ions that regulate metabolic processes are called _______. Please reason step by step, and put your final answer within \boxed{}. | electrolytes |
user | Evaluate the integral $$\int\limits_0^1\frac{\ln x}{x^{a}(1-x)^{b}}\,\mathrm{d}x,$$ where $a,b \in (0,1)$. Please reason step by step, and put your final answer within \boxed{}. | \frac{\partial}{\partial a}\mathbf B(1-a,1-b) |
user | In 1892 George W. G. Ferris designed a carnival ride in the shape of a large wheel. This Ferris wheel had a diameter of {eq}76 \ m
{/eq} and rotated one revolution every {eq}20 \ min
{/eq}. What was the magnitude of the acceleration that riders experienced? Please reason step by step, and put your final answer within... | 0.001 m/s^2 |
user | Evaluate the integral by reversing the order of integration.
{eq}\int_{0}^{2} \int_{x}^{2} \; 5e^{\frac{x}{y}} \, \mathrm{d}y \, \mathrm{d}x{/eq} Please reason step by step, and put your final answer within \boxed{}. | 10e - 10 |
user | An astronaut is connected to her spacecraft by a 27-m-long tether cord as she and the spacecraft orbit Earth in a circular path at a speed of 3.310{eq}^3
{/eq} m/s. At one instant, the voltage measured between the ends of a wire embedded in the cord is measured to be 0.40 V. Assume the long dimension of the cord is p... | 4.489e-6 T |
user | An engine removes 260 J from a reservoir at 385 K and exhausts 210 J to a reservoir at 200 K. How much more work could be done if the engine were an ideal (Carnot) reversible engine? Please reason step by step, and put your final answer within \boxed{}. | 124.8 J |
user | A ball of negligible size is placed on a sphere of the same mass. The ball is slightly pushed after some time the ball makes an angle of {eq}30^{\circ}{/eq} with the vertical speed of the sphere find the velocity of the sphere? Please reason step by step, and put your final answer within \boxed{}. | 0.89m/s |
user | Let $f\colon X\to\overline{\mathbb{R}}_{\geq 0}$ be a measurable function. Show that $$ \int f\, d\mu=\int\limits_0^{\infty}\mu(\left\{x\in X: f(x)>t\right\})\, dt. $$
The questioner proposes to approximate $f$ by the functions $$ f_n:=2^{-n}\sum\limits_{k=1}^{\infty}\chi_{\left\{x\in X: f(x)> k/2^n\right\}}. $$
The q... | \int_Xf(y)\;d\mu(y) |
user | At a certain time a particle had a speed of {eq}59\ m/s
{/eq} in the positive x direction, and {eq}3.1\ s
{/eq} later its speed was {eq}84\ m/s
{/eq} in the opposite direction. What was the average acceleration of the particle during this {eq}3.1\ s
{/eq} interval? Please reason step by step, and put your final ans... | 46 m/s^2 |
user | Find the point on the plane 4x + 3y + z = 26 that is closest to the origin by minimizing the square of the distance. Please reason step by step, and put your final answer within \boxed{}. | (4, 3, 1) |
user | How many grams of ethyl alcohol are needed to produce 1 L of a 2 mM solution, if 1 mole = 46 g of ethyl alcohol? Please reason step by step, and put your final answer within \boxed{}. | 0.09214 g |
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