PRIME
Collection
PRIME data & models
•
4 items
•
Updated
•
1
role
stringclasses 1
value | content
stringlengths 80
4.77k
| ground_truth
stringlengths 0
1.45k
|
|---|---|---|
user
|
When methane gas is burned, carbon dioxide is released into the atmosphere. How much {eq}CO_2(g)
{/eq} is released if 78.5 L of methane is burned?
{eq}CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O
{/eq} Please reason step by step, and put your final answer within \boxed{}.
|
78.5 L
|
user
|
How do you differentiate #sqrt(sin^3(1/x^2) # using the chain rule? Please reason step by step, and put your final answer within \boxed{}.
|
#-3x^-3[(sin(x^-2))^3]^(-1/2)(sin(x^-2))^2(cos(x^-2))#
|
user
|
The chief erosive agent on the moon is what? Please reason step by step, and put your final answer within \boxed{}.
|
micrometeorites
|
user
|
If 2.68 g of sodium sulfide reacts with 10.6 g of copper (I) nitrate, how much of the product containing the transition metal is produced? Please reason step by step, and put your final answer within \boxed{}.
|
3.28 g
|
user
|
A converging lens with focal length 12 cm and a diverging lens with focal length 15 cm are located 24 cm apart. An object is placed 10 cm to the left of the diverging lens. A plane mirror is placed 25 cm to the right of the converging lens.
Where is the image? Please reason step by step, and put your final answer within \boxed{}.
|
5 cm behind the mirror
|
user
|
The light-emitting decay of excited mercury atoms is first-order, with a rate constant of 1.65E6 1/s. A sample contains 5.2E-6 M of excited mercury atoms.
Find the concentration after 2.5E-6 s. Please reason step by step, and put your final answer within \boxed{}.
|
8.41E-8 M
|
user
|
A 0.96 millimole sample of {eq}Na_2CO_3
{/eq} gave 0.9 millimoles of {eq}CO_2
{/eq} gas. If a 0.376 g of pure {eq}Na_2CO_3
{/eq} was reacted with excess acid, what volume of gas will be measured on this apparatus? Please reason step by step, and put your final answer within \boxed{}.
|
74.53 mL
|
user
|
A random sample of 61 students was asked how much they spent on classroom textbooks this semester. The sample standard deviation was found to be s = $28.70. How many more students should be included in the sample to be 99% sure the sample mean x is within $7 of the population mean {eq}\mu
{/eq} for all students at this college?
(a) 0 .
(b) 65.
(c) 51.
(d) 4.
(e) 112. Please reason step by step, and put your final answer within \boxed{}.
|
E
|
user
|
A crude oil burned in electrical generating plant contains about {eq}1.2 \ \%
{/eq} sulfur by mass. When the oil burns, the sulfur forms sulfur dioxide gas: {eq}S (s) + O_2 (g) \longrightarrow SO_2 (g).
{/eq}
(a) How many litres of {eq}SO_2 \ (d = 2.60 \ g / L)
{/eq} are produced when {eq}1.00 \times 10^4 \ kg
{/eq} of oil burns at the same temperature and pressure ? Please reason step by step, and put your final answer within \boxed{}.
|
9.23e+04 L
|
user
|
Let $C \subset \mathbb{P}_\mathbb{C}^2$ be a projective curve, and $p \in C$ a point of multiplicity $m$. If $\mathcal{O}_p(C)$ is the local ring of $C$ at $p$, and $\mathfrak{m} \subset \mathcal{O}_p(C)$ is its maximal ideal, what is $\text{dim}_\mathbb{C} \mathfrak{m}^k/\mathfrak{m}^{k+1}$? Please reason step by step, and put your final answer within \boxed{}.
|
$k+1$
|
user
|
I have the following objective function: $$\sum_{i=1}^n a_i^2 -\lambda \sum_{i=1}^n a_i - \mu\sum_{i=1}^n a_ix_i, $$ where $\lambda$ and $\mu$ are lagrange multipliers. Differentiating with respect to $a_i, \lambda$, and $\mu$ yield that $2a_i -\lambda -\mu x_i$ and $\sum_{i=1}^n a_i$ and $\sum_{i=1}^n a_ix_i$, respectively. If we let these equations equal to zero, the second and third equations are just two constraints. How do I use the first equation to find $a_i$? Please reason step by step, and put your final answer within \boxed{}.
|
$a_i=\frac{x_i-\bar{x}}{\sum (x_i-\bar{x})^2}$
|
user
|
Find the derivative of the function and evaluate the derivative at the given of a: {eq}h(x)=x^{\sqrt x}; \ a=16{/eq} Please reason step by step, and put your final answer within \boxed{}.
|
39097.047
|
user
|
Is there a closed-form solution for the sum $\sum_{k=0}^{n}B_{n,k}(g'(t),g''(t),\cdots)$, where $B_{n,k}$ are the Bell polynomials of the second kind? Please reason step by step, and put your final answer within \boxed{}.
|
$\frac{d^{n}}{dt^{n}}e^{g(t)}=e^{g(t)}B_{n}(g'(t),g''(t),...)$
|
user
|
Find the average value of the function f(t) = 2^{t/10} + \frac{1}{t} + \sin(2t - 3) on the interval [1, 9]. Please reason step by step, and put your final answer within \boxed{}.
|
\frac{1}{8}\left [ \frac{10}{ \ln 2}(2^{9/10}-2^{1/10})+\ln 9+\frac{1}{2}(\cos 1-\cos 15) \right ]
|
user
|
I want to solve $y'' = y^3 -y y'$ with boundary conditions $y(1) = 1/2$ and $y(2) = 1/3$ but not sure how to start. By the hint $(y'+\frac{y^2}{2})'=y^3$ $y' + \frac{y^2}{2} = \frac{y^4}{4} + c1$ $\frac{dy}{dx} = \frac{y^4}{4} - \frac{y^2}{2} + c1$ Integrate $\frac{dy}{\frac{y^4}{4} - \frac{y^2}{2} + c1} = dx$. Integration looks very messy. How do I proceed to solve the equation with the given boundary conditions, knowing that the answer is $y = \frac{1}{(x+1)}$? Please reason step by step, and put your final answer within \boxed{}.
|
$y = \frac{1}{(x+1)}$
|
user
|
What are the intervals of increasing and decreasing concavity with the function f(x)= \frac {1}{1+x^2} Please reason step by step, and put your final answer within \boxed{}.
|
The function is concave up when {eq}x\epsilon \left ( \frac{1}{\sqrt{3}}, \infty \right ){/eq} and concave down when {eq}x\epsilon (-\infty, \frac{1}{\sqrt{3}}){/eq}
|
user
|
Find the value of the integral: $$\int_{\frac{1}{4}}^4\frac{(x+1)\arctan x}{x\sqrt{x^2+1}} dx$$ Please reason step by step, and put your final answer within \boxed{}.
|
$\frac{\pi}{4}\int_{1/4}^4 \frac{x+1}{x\sqrt{x^2+1}}dx$ or $I=\frac{\pi}{4}\int_{\frac{1}{4}}^4\frac{t+1}{t\sqrt{t^2+1}}dt$
|
user
|
25 milliliters of the tartaric acid solution is titrated against 31 milliliters of {eq}NH_4OH.
{/eq} What is the normality of the {eq}NH_4OH?
{/eq} Please reason step by step, and put your final answer within \boxed{}.
|
0.81 N NH4OH
|
user
|
We need to show that the tension in the rope, T, is = to 2Mgcos(theta) I've taken Moments about A but keep getting T = (1/2)Mgcos(theta). I've ignored the moments at C as they are in equilibrium already I'm assuming. Meaning (1.5a)Rc = (1.5a)2Mgcos(theta). Including them in my calculation doesn't make a difference. Any help would be appreciated. Please reason step by step, and put your final answer within \boxed{}.
|
T = 2Mgcos(theta)
|
user
|
Describe how carbon dioxide is transported in the blood. Where is it distributed in the blood and in what proportions? Please reason step by step, and put your final answer within \boxed{}.
|
1. Dissolved in blood plasma (5-7%)
2. Bound to hemoglobin as carbaminohemoglobin (10%)
3. Carried as bicarbonate ions (85%)
|
user
|
I would like help solving the next differential equation. $$y''=y'^2+1$$ I tried subsitiuting $p=y'$ and got to $\frac{1}{2}\ln|p^2+1|=\frac{1}{2}\ln|y|+\ln|c|$ but I don't know how to continue and maybe made a mistake. any ideas? Please reason step by step, and put your final answer within \boxed{}.
|
y = -ln(cos(x + C1)) + C2
|
user
|
Apply Green's Theorem to evaluate the integral {eq}\oint_{C} (8y + x)\ dx + (y + 5x)\ dy {/eq} where {eq}C {/eq} is the circle {eq}(x - 7)^2 + (y - 6)^2 = 5 {/eq}, oriented counterclockwise. Please reason step by step, and put your final answer within \boxed{}.
|
-15π
|
user
|
A DNA strand of sequence 5'-ATCAGC-3' will be complementary to a DNA strand with the sequence.
A) 3'-GCUGAU-5'
B) 3'-ATCACG-5'
C) 5'-AUCAGC-3'
D) 5'-GCTGAT-3'
E. 5'-TAGTCG-3' Please reason step by step, and put your final answer within \boxed{}.
|
E
|
user
|
If I have 2.4 moles of gas held at a temperature of 97 C and in a container with a volume of 45 liters, what is the pressure of the gas in torr? Please reason step by step, and put your final answer within \boxed{}.
|
1.2e3 torr
|
user
|
Suppose that we wanted to estimate the true average number of eggs a queen bee lays with 95 percent confidence. The margin of error we are willing to accept is 0.5. Suppose we also know that s is about 10. At minimum, what sample size should we use? Please reason step by step, and put your final answer within \boxed{}.
|
1537
|
user
|
What is the orbital radius of the electron in the n = 4 state of hydrogen? Please reason step by step, and put your final answer within \boxed{}.
|
8.48*10^-10 m
|
user
|
Suppose that the marginal revenue for a product is given by
{eq}\overline {MR} = \frac {-22}{(2x + 1)^2} + 22
{/eq}
where x is the number of units and revenue is in dollars.
Find the total revenue, R(x). Please reason step by step, and put your final answer within \boxed{}.
|
R(x) = 22 {\frac {1}{2(2x + 1)} + x} + C
|
user
|
A stationary block explodes into two pieces L and R that slide across a frictionless floor and then into regions with friction, where they stop. Piece L, with a mass of 2.0 kg, encounters a coefficient of kinetic friction {eq}\mu_L
{/eq} = 0.40 and slides to a stop in distance dL = 0.15 m. Piece R encounters a coefficient of kinetic friction {eq}\mu_R
{/eq} = 0.50 and slides to a stop in distance dR = 0.20 m. What was the mass of the original block? The momentum of L and R are equal. Please reason step by step, and put your final answer within \boxed{}.
|
3.5 kg
|
user
|
How can I justify the exponential decay behavior of $p(n)$ in the system
$$\begin{aligned} v(n) = 0.6 \cdot v(n-1) \\ p(n) = 0.13 \cdot v(n) + 0.87 \cdot p(n-1) + 25 \end{aligned}$$
with initial values $v(0) \approx 1441.67$, $p(0) = 3000$ and $p(1) = 3500$, given that $v(n)$ clearly decays exponentially, and while expanding $p(n)$ shows exponential terms, the overall expression is too complex to easily determine if the decay is a "clean" exponential decay? Please reason step by step, and put your final answer within \boxed{}.
|
$\begin{aligned}v(n)&=0.6^n*v(0) \\ p(n)&=0.87^n*(25+p(0))+v(0)(0.419*0.87^n-0.289*0.6^n)\end{aligned}$
|
user
|
I'm going on a holiday to Mumbai. I call my friend who lives there and ask him if it is raining. He has a one in third chance of lying. He assures me that it is raining. What is the probability of it actually raining? Why can't I disregard the prior probability of it raining on a given day, and what is the correct approach to solving this problem? Please reason step by step, and put your final answer within \boxed{}.
|
2p / (1+p)
|
user
|
How can I prove \begin{align*} \frac{2j+1}{z^{n+1}} \sum_{n=0}^{2j} [\frac{(2j)!}{(2j- n)! n!}]^{\frac{1}{2}}f^*_n \int_{0}^{\infty} \frac{(t)^n}{(1+t)^{2j+2}}dt \end{align*} is equal to $=$ \begin{align*} \sum_{n=0}^{2j} [\frac{((2j-n)!n!)}{(2j!)}]^{\frac{1}{2}}f^*_n z^{-n-1} \end{align*} where (j=1/2,1,3/2,2....),(n is an integer ≥ 0 ) and f^*_n is a complex conjogate of f^_n(complex state)? Please reason step by step, and put your final answer within \boxed{}.
|
\int_0^\infty\frac{t^n}{(1+t)^{2j+2}}\,dt = \frac{(2j-n)!n!}{(2j+1)!}
|
user
|
How do you use a Riemann sum to calculate a definite integral? Please reason step by step, and put your final answer within \boxed{}.
|
\displaystyle\int\limits_a^b f (x)\;dx = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n f ({{x^*}_i})\Delta {x_i} \quad \quad \quad {\rm where} \qquad \qquad \Delta {x_i} = {x_i} - {x_{i - 1}}
|
user
|
How to solve $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$ together with the initial condition $X(0) = X_0$? Please reason step by step, and put your final answer within \boxed{}.
|
$X_t = U_t^{-1} \left[ x_0 + \int_0^t \! \left(c(s) - e(s)f(s) \right)U_s\mathrm{d}s + \int_0^t \!e(s)U_s\mathrm{d}W_s \right]$ where $U_t^{-1}= \frac{1}{U_t}=\exp \left( -\int_0^t \! \left(d(s)- \tfrac 12 f^2(s)\right)\mathrm{d}s - \int_0^t \! f(s)\mathrm{d}W_s \right)$
|
user
|
There is a ball of radius $a$ and a point $R$ inside this ball $(|R| <a)$. Calculate the integral: $$T\left(R\right)=\int_{r<a}^{ }\frac{d^{3}r}{\left|R-r\right|^{2}}$$ Please reason step by step, and put your final answer within \boxed{}.
|
\(\pi \left(\frac{a^2-R^2}R\ln\frac{a+R}{a-R}+2a\right)\)
|
user
|
A rounded knob that articulates with another bone is called a Please reason step by step, and put your final answer within \boxed{}.
|
condyle
|
user
|
Calculate the reactance of a {eq}3.6\text{ mH}
{/eq} inductor at a frequency of {eq}5\text{ kHz}
{/eq}. Please reason step by step, and put your final answer within \boxed{}.
|
113.1Ω
|
user
|
A stock car race is held on a circular track that is approximately 2.2 km long, and the turns are banked at an angle of 15.4. It is currently possible for cars to travel through the turns at a speed of about 51.85664 m/s. Assuming that these cars are on the verge of slipping into the wall of the race track (since they are racing!), find the coefficient of static friction between the tires and the track. Please reason step by step, and put your final answer within \boxed{}.
|
0.418
|
user
|
I'm reading Concrete Mathematics by Graham, Knuth, Patashnik. I found that for every integer $n$, this holds: $$n = \lceil n/m \rceil + \lceil (n-1)/m \rceil + \cdots + \lceil (n-m+1)/m \rceil$$ Why is this true? Is there any name for this property? Please reason step by step, and put your final answer within \boxed{}.
|
n
|
user
|
Is $$n\left(1-(1-(2r+1)/N)^{n-1}\right) \choose 2$$ correct for the expected number of pairs of $n$ people that share their birthday within $r$ days of each other, given that the original birthday problem's expected number of people is $n\left(1-(1-1/N)^{n-1}\right)$? Please reason step by step, and put your final answer within \boxed{}.
|
n(1-(1-(2r+1)/N)^(n-1))
|
user
|
A balloon filled with helium gas at 20 degrees Celsius occupies 6.91 L at 1.00 atm. The balloon is immersed in liquid nitrogen at -196 degrees Celsius, while the pressure is raised to 5.20 atm. What is the volume of the balloon in the liquid nitrogen? Please reason step by step, and put your final answer within \boxed{}.
|
0.36 L
|
user
|
Calculate the atomic mass of element X if it has 2 naturally occurring isotopes with the following masses and natural abundances.
X-45: mass = 44.8776 amu, abundance = 32.88%
X-47: mass = 46.9443 amu, abundance = 67.12% Please reason step by step, and put your final answer within \boxed{}.
|
46.26 amu
|
user
|
There are 10 men and 10 women. How many ways are there to arrange these 20 people at a round table with 25 seats? (Rotations are considered the same arrangement.) Please reason step by step, and put your final answer within \boxed{}.
|
6.46e+21
|
user
|
How do I express an nth order polynomial in terms of the Chebyshev terms of the first kind? In other words, how do I express f(x) = $a_0$+$a_1$x+...+$a_nx^n$ in terms of $b_0T_0+b_1T_1+...+b_nT_n$? Please reason step by step, and put your final answer within \boxed{}.
|
$x^k = \frac{1}{2^{k-1}}\left(T_k(x) + {k \choose 1}T_{k-2}(x) + {k \choose 2}T_{k-4}(x) + \cdots \right)$
|
user
|
Find the interval and radius of convergence of the power series
{eq}\displaystyle \sum^{\infty}_{k = 1} \frac {(x-3)^k}{k^2}
{/eq} Please reason step by step, and put your final answer within \boxed{}.
|
radius of convergence is 3 and the interval of convergence is 2 <= x <= 4
|
user
|
Suppose {eq}F(x,y) = 3y\hat i + 6xy\hat j{/eq}. Use Green's Theorem to calculate the line integral of F around the perimeter of a circle C of radius 3 centered at the origin and oriented counter-clockwise. Please reason step by step, and put your final answer within \boxed{}.
|
-27π
|
user
|
What is the probability a certain tire will last between 57,690 miles and 58,020 miles, given the tread life is normally distributed with a mean of 60,000 miles and a standard deviation of 1100 miles? Please reason step by step, and put your final answer within \boxed{}.
|
0.018
|
user
|
Find the maximum speed {eq}v_\mathrm{max}
{/eq} of a simple pendulum bob in terms of {eq}g
{/eq}, the length {eq}L
{/eq}, and the initial angle of swing {eq}\theta_0
{/eq}. Use the small angle approximation. Please reason step by step, and put your final answer within \boxed{}.
|
v_m = \sqrt { g L \theta_0^2 }
|
user
|
Compute the rank of
$\left(
\begin{array}{cccc}
-\frac{125}{16} & -\frac{3}{16} & -\frac{1}{16} & \frac{79}{8} \\
\end{array}
\right)$. Please reason step by step, and put your final answer within \boxed{}.
|
1
|
user
|
Each step in the process below has a 70.0% yield.
{eq}CH_4 + 4Cl_2 \to CCl_4 + 4HCl
{/eq}
{eq}CCl_4 + 2HF \to CCl_2F_2 + 2HCl
{/eq}
The {eq}CCl_4
{/eq} formed in the first step is used as a reactant in the second step. If 7.00 moles of {eq}CH_4
{/eq} reacts, what is the total amount of {eq}HCl
{/eq} produced? Assume that {eq}Cl_2
{/eq} and {eq}HF
{/eq} are present in excess. Please reason step by step, and put your final answer within \boxed{}.
|
29.4 mol
|
user
|
Write the balanced ionic equation for the neutralization reaction between {eq}\rm NH_3(aq)
{/eq} and {eq}\rm H_2SO_4(aq)
{/eq}. Please reason step by step, and put your final answer within \boxed{}.
|
2NH_3 + 2H^+ → 2NH_4^+
|
user
|
Let $X$ be a random variable with Normal distribution: $N(m,\sigma^2)$. Let $\eta$ be a constant. Now, let $M=\min(X,\eta)$. What is the expectation and variance of $M$? Please reason step by step, and put your final answer within \boxed{}.
|
$\operatorname{E}[\min(X , \eta)] = \eta + (\mu-\eta) \, \Phi\left(\tfrac{\eta - \mu}{\sigma}\right) - \frac{\sigma}{\sqrt{2\pi}} e^{-(\eta - \mu)^2/(2\sigma^2)}$
|
user
|
Evaluate the indefinite integral: ∫ (1 + 4/x)^2 dx Please reason step by step, and put your final answer within \boxed{}.
|
x + 8ln(x) - 16/x + C
|
user
|
If you assume that an atom is a hard sphere, you can estimate the average volume of space it occupies using basic geometry.
Estimate the volume, V, of a sodium atom ( Na ) using its metallic radius of 186. Please reason step by step, and put your final answer within \boxed{}.
|
2.7e-29 m^3
|
user
|
What monomer is made up of glycerol and another molecule? Please reason step by step, and put your final answer within \boxed{}.
|
Fat/Oil
|
user
|
- The preciptation of iron(II) oxalate dihydrate Write the net ionic equation for the precipitation of iron(II) oxalate dihydrate when oxalic acid is added to a solution of iron(II) ions. Include phases in your equation. Remember to use the center dot (from the formatting menu) for the waters of hydration. Please reason step by step, and put your final answer within \boxed{}.
|
Fe^(2+)(aq) + C2O4^(2-)(aq) + 2H2O(l) -> FeC2O4⋅2H2O(s)
|
user
|
A mixture of 2.00 grams of {eq}H_2
{/eq} and 16.00 grams of {eq}O_2
{/eq} is placed in a 2.00 L flask at 35.0 degrees Celsius. Assuming each gas behaves as an ideal gas, calculate the partial pressure of oxygen gas. Please reason step by step, and put your final answer within \boxed{}.
|
6.32 atm
|
user
|
an airplane tries to fly due north at 100 m/s but a wind is blowing from the west at 30 m/s. what heading (angle and direction) should the plane take to go due north in spite of the wind? Please reason step by step, and put your final answer within \boxed{}.
|
78.1° north from the west direction
|
user
|
Calculate a 99% confidence interval for the population mean, given a sample of 50 units with a mean of 75 pounds and a standard deviation of 10 pounds. The weight of the product is measured in pounds. Please reason step by step, and put your final answer within \boxed{}.
|
(71.21, 78.79)
|
user
|
An object has a mass of #4 kg#. The object's kinetic energy uniformly changes from #16 KJ# to # 66KJ# over #t in [0, 6 s]#. What is the average speed of the object? Please reason step by step, and put your final answer within \boxed{}.
|
140.8ms^-1
|
user
|
Solid potassium reacts with oxygen gas to produce solid potassium oxide. balanced chemical equation? Please reason step by step, and put your final answer within \boxed{}.
|
4K(s) + O2(g) -> 2K2O(s)
|
user
|
A long chain fatty acid generates more ATP than a molecule of glucose though which process? Please reason step by step, and put your final answer within \boxed{}.
|
Beta oxidation
|
user
|
A rod of length L = 4.0 m with uniform charge of 7.9 nC/m is oriented along the y axis.
What is the electric field vector at the location P whose coordinates are (0, -6.0 m) ?
Given (E = N/C) Please reason step by step, and put your final answer within \boxed{}.
|
-227N/C i
|
user
|
Calculate the mass of {eq}CO_2
{/eq} produced when {eq}2CH_3OH
{/eq} mixed with {eq}O_2
{/eq}. Given that {eq}2CH_3OH
{/eq} is {eq}4O
{/eq} grams and {eq}O_2
{/eq} is 46 grams. Please reason step by step, and put your final answer within \boxed{}.
|
54.94 g
|
user
|
Evaluate {eq}\displaystyle
\lim_{ x \to \infty} x^3 e^{\frac{-x}{5}}
{/eq}
{eq}\displaystyle
\circ 1 \\\circ e \\ \circ -e \\ \circ 0
{/eq} Please reason step by step, and put your final answer within \boxed{}.
|
D
|
user
|
The stratospheric ozone {eq}(O_3)
{/eq} layer helps to protect us from harmful ultraviolet radiation. It does so by absorbing ultraviolet and falling apart into an {eq}O_2
{/eq} molecule and an oxygen atom, a process known as photodissociation.
$$O_3(g) \rightarrow O_2(g) + O(g)
$$
The energy that is needed to cause the complete dissociation of 144.0 grams of ozone is 300.0 kJ. What is the maximum wavelength (in nm) a photon can have if it is to possess sufficient energy to cause this dissociation? Please reason step by step, and put your final answer within \boxed{}.
|
1197 nm
|
user
|
Some trolls have one eye (EE, Ee) while others have two (ee). Two heterozygous one-eyed trolls are crossed. What is the expected phenotype ratio? Please reason step by step, and put your final answer within \boxed{}.
|
3:1 one eye: two eyes
|
user
|
Two equal charges, 49.2 micro Coulomb each, are separated by 5 cm. Find the force between them. Please reason step by step, and put your final answer within \boxed{}.
|
8714.304 N
|
user
|
So we're given this probability function $$p(x) = e^{{-a(x-b)}^2}$$ and we have to find the expectation value of $x$ which would be $$ \left<x\right> = \int_{-\infty}^{\infty}p(x)^*x p(x) = \int_{-\infty}^{\infty}xe^{{-2a(x-b)}^2} $$ The problem for me is when I do integration by parts I need the indefinite integral of $p(x)$ to put into the integration formula. All proofs I've found use trig identities that I'm not sure work with more than just $x^2$ in the exponent. Anyone willing to explain how to find the indefinite integral of $p(x)$? Please reason step by step, and put your final answer within \boxed{}.
|
b
|
user
|
{eq}\rm 100\ ml
{/eq} solution made up {eq}\rm 1.0\ M\ NH_3
{/eq} and {eq}\rm (NH_4)_2SO_4
{/eq} have been added with {eq}\rm 100\ ml
{/eq} of {eq}\rm 0.1 \ M\ HCl
{/eq}. How many moles are present in the solution before and after? Given the pH of the solution is {eq}9.26
{/eq}. Please reason step by step, and put your final answer within \boxed{}.
|
Before reaction:
moles of HCl = 0.010 mol
moles of ammonium = 0.20 mol
moles of ammonia = 0.10 mol
After reaction:
moles of ammonium = 0.21 mol
moles of ammonia = 0.09 mol
|
user
|
What is a solution to the differential equation dy/dx=e^(x+y)? Please reason step by step, and put your final answer within \boxed{}.
|
y = -ln(-e^x + C)
|
user
|
How many of each ion are contained within a unit cell of CsCl Please reason step by step, and put your final answer within \boxed{}.
|
1 Cs ion and 1 Cl ion
|
user
|
Find the curvature of the cycloid {eq}x = a - sin(x), y = 1 - cos(x){/eq} at the top of one of its arches, using the formula {eq}k=\frac{|xy-xy|}{[x^{2}+y^{2} ]^{\frac{3}{2}}}{/eq}, where the dots indicate derivatives with respect to {eq}t{/eq}, so {eq}f = \frac{dx}{dt}{/eq}. Please reason step by step, and put your final answer within \boxed{}.
|
1
|
user
|
Let $f(z)$ and $g(z)$ be analytic in a region A and let $g'(z) \neq 0$ for all $z \in A$. Let g(z) be one to one and let $\gamma$ be a closed curve in A. Show that $$ f(z_0)I(\gamma;z_0)=\frac {g'(z_0)}{2\pi i}\int_{\gamma} \frac {f(z)}{g(z)-g(z_0)}dz. $$ Please reason step by step, and put your final answer within \boxed{}.
|
$I(\gamma;z_0) \frac{f(z_0)}{g'(z_0)}$
|
user
|
Let {eq}C_1{/eq} be the arc of the intersection of the cone {eq}z = 6- \sqrt {x^2 + y^2 }{/eq} with the plane {eq}z=2{/eq}, from {eq}(4,0,2){/eq} to {eq}(0,4,2){/eq}. Let {eq}C_2{/eq} be the arc of intersection of the same cone with the plane {eq}y=4{/eq}, from {eq}(0,4,2 ){/eq} to {eq}(2 \sqrt 5 , 4,0 ){/eq}. Find the work done by the vector field {eq}F(x,y,z) = <x,y,z>{/eq} along the curve {eq}C = C_1 \cup C_2{/eq}. Please reason step by step, and put your final answer within \boxed{}.
|
8
|
user
|
Let F (x, y) = -yi+xj/x^2+y^2 and let C be the circle r (t) = (cos t) i + (sin t)j . Compute dq/dx. Please reason step by step, and put your final answer within \boxed{}.
|
(y^2-x^2)/(x^2+y^2)^2
|
user
|
An out-of-control truck with a mass of 5000 kg is traveling at 35.0 m/s when it starts descending a steep (15 degree) incline. The incline is icy, so the coefficient of friction is only 0.30. Use the work-energy theorem to determine how far the truck will skid (assuming it locks its brakes and skids the whole way) before coming to rest. Please reason step by step, and put your final answer within \boxed{}.
|
2037.05 m
|
user
|
If a rubber tire having a weight of 18.0lbs requires a force of 38.0N to keep it in uniform motion on a level highway, find the coefficient of kinetic friction. Please reason step by step, and put your final answer within \boxed{}.
|
0.47
|
user
|
If an epithelium has cells that are thin and flat, then the shape of the cell will be:
a) cuboidal
b) columnar
c) squamous
d) striated Please reason step by step, and put your final answer within \boxed{}.
|
c
|
user
|
Find the ratio of speeds of an electron and a negative hydrogen ion (one having an extra electron) accelerated through the same voltage, assuming non-relativistic final speeds. Take the mass of the hydrogen ion to be {eq}1.67 \times10^{27}
{/eq} kg. Please reason step by step, and put your final answer within \boxed{}.
|
42.83
|
user
|
An electron starts from rest 6.00 cm from the center of a uniformly charged insulating sphere of radius 5.00 cm and total charge 1.14 nC. What is the speed of the electron when it reaches the surface of the sphere? Please reason step by step, and put your final answer within \boxed{}.
|
3.4e6 m/s
|
user
|
Find the derivative of the function {eq}y = \displaystyle \ln \left|\frac{\cos x}{\cos x - 1}\right|{/eq}. Please reason step by step, and put your final answer within \boxed{}.
|
\frac{-\sin \left(2x\right)\left(\cos \left(x\right)-1\right)+2\cos ^2\left(x\right)\sin \left(x\right)}{2\cos ^2\left(x\right)\left(\cos \left(x\right)-1\right)}
|
user
|
What is an incomplete fracture in which the fracture is apparent only on convex surfaces? Please reason step by step, and put your final answer within \boxed{}.
|
greenstick fracture
|
user
|
Find the image of $\gamma = \{z : \mathbb{Re}(z) = \mathbb{Im}(z)\}$ under $f(z) = \frac{z+1}{z-1}$. Find the center and radius of the resulting circle. Please reason step by step, and put your final answer within \boxed{}.
|
$u^2+2uv+3v^2=1$
|
user
|
Differentiate the following function:
$(-4 x-1) \log (-8 x-5)$ Please reason step by step, and put your final answer within \boxed{}.
|
\frac{4 (2+8 x+(5+8 x) \log (-5-8 x))}{-5-8 x}
|
user
|
Evaluate the integral: {eq}\displaystyle \int [(1-\sin^2 \theta) \cdot \csc \theta] \ d \theta{/eq} Please reason step by step, and put your final answer within \boxed{}.
|
\cos (\theta) - \ln |\csc (\theta) + \cot (\theta)| + C
|
user
|
A system containing an ideal gas at a constant pressure of 1.22 x {eq}10^5
{/eq} Pa gains 2,140 J of heat. During the process, the internal energy of the system increases by 2,320 J. What is the change in volume of the gas? Please reason step by step, and put your final answer within \boxed{}.
|
-1.475e-3 m^3
|
user
|
Is it possible to express $i$ as a linear combination of the basis $\{1, \sqrt[4]2, \sqrt[4]2^2, \sqrt[4]2^3\}$ with coefficients in $\mathbb{Q}$, and if not, is there an error in the reasoning that leads to the conclusion that the degree of the extension from $\mathbb{Q}$ to $\mathbb{Q}(i, \sqrt[4]2)$ is 4? Please reason step by step, and put your final answer within \boxed{}.
|
8
|
user
|
Suppose the cost in dollars to make x oboe reeds is given by C(x) = 5 log_2 x + 10. Find the marginal average cost when 10 reeds are sold. Please reason step by step, and put your final answer within \boxed{}.
|
-0.194
|
user
|
Jack and Jill are thinking about starting a family and they come to you with a question about a rare recessive genetic disease called Alpers Syndrome. Alpers causes progressive neurodegeneration in infants and children and is usually lethal within the first 10 years of life. Jack's grandfather had a brother die of Alpers Syndrome, while Jill has no family history of the disease.
If the frequency of heterozygotes for Alpers is 1 in 2000 in the general population, what is the chance that Jack and Jill will have a child with Alpers Syndrome? Please reason step by step, and put your final answer within \boxed{}.
|
0.0000156
|
user
|
Consider a region $V$ bounded by the paraboloid $z=5-4x^2-4y^2$ and the $xy$-plane. The surface integral for the vector field $$\vec{F}=\bigtriangledown\times \vec{G}=2\vec{i}+2y^2\vec{j}+z\vec{k}$$ over the circle in the $xy$-plane is 15. What is the value for the surface integral over the paraboloid? Please reason step by step, and put your final answer within \boxed{}.
|
-15
|
user
|
For a standard normal distribution, what are the {eq}z_{\alpha/2} values that correspond with a confidence interval of {eq}80 \%{/eq}? Please reason step by step, and put your final answer within \boxed{}.
|
1.28
|
user
|
How many grams of {eq}NH_3
{/eq} can be produced from 4.78 mol of {eq}N_2 \text{ and excess } H_2?
{/eq} Please reason step by step, and put your final answer within \boxed{}.
|
9.56 mol NH_3
|
user
|
Approximate {eq}e^9{/eq} using a Taylor polynomial of {eq}\displaystyle f (x) = e^x{/eq} centered at {eq}a = 0{/eq}. Please reason step by step, and put your final answer within \boxed{}.
|
8103.083928
|
user
|
Is there a power series expansion for the inverse of a matrix that has the form $(\mathbf{K}^T\mathbf{K}+\mathbf{A})^{−1}$, where $\mathbf{K}_{m \times n}$ and $\mathbf{A}_{n \times n}$ is not invertible, but $\mathbf{K}^T\mathbf{K}+\mathbf{A}$ is? Please reason step by step, and put your final answer within \boxed{}.
|
$(K^TK + A)^{-1} = \Big(\sum_{k=0}^{\infty} \big((K^TK)^{-1}A\big)^k \Big)(K^TK)^{-1}$ if $K^TK$ is invertible and the spectral radius $\rho\big((K^TK)^{-1}A\big) < 1$
|
user
|
Show that for $k\ge1$, \begin{align} \\&\quad(-1)^k\sum_{n=0}^{\infty}{2^{n+1}(2k-1)!!\over {2n\choose n}(2n+1)(2n+3)\cdots(2n+2k+1)}\\[10pt]&=\pi-4\left(1-{1\over 3}+{1\over 5}-{1\over 7}+\cdots+{1\over 2k-1}\right). \end{align} Please reason step by step, and put your final answer within \boxed{}.
|
$S_k=(-1)^k\left(\psi \left(\frac{k}{2}+\frac{3}{4}\right)-\psi\left(\frac{k}{2}+\frac{1}{4}\right)\right)$
|
user
|
A water tank is filled to a depth of 10 m, and the bottom of the tank is 20 m above ground. A water-filled hose that is 2.0 cm in diameter extends from the bottom of the tank to the ground, but no water is flowing in this hose. The water pressure at ground level in the hose is closest to which of the following values? The density of water is 1000 {eq}kg/m^3.
{/eq} Please reason step by step, and put your final answer within \boxed{}.
|
395.325 kPa
|
user
|
Consider the normed linear space ($ \mathbb C^2$, $ $|| ||$_\infty$). $f$ is a linear functional on the space given by, $f(x,y)=ax+by$, for some constant a,b. Given that $||f|| \leq |a|+|b|$, how can I prove that $||f|| = |a|+|b|$? Please reason step by step, and put your final answer within \boxed{}.
|
||f|| = |a| + |b|
|
user
|
A laser pulse produces 2.098 kJ of energy. It was experimentally determined that the pulse contains {eq}7.50 \times 10^{22}
{/eq} photons. Determine the wavelength (in meters) emitted by one photon? Please reason step by step, and put your final answer within \boxed{}.
|
7.107e-6 m
|
user
|
For what values of x is g continuous, where
g(x) =
\begin{cases} 0 &\text{if}\ x\ \text{is rational}\\ x & \text{if}\ x\ \text{is irrational} \end{cases} Please reason step by step, and put your final answer within \boxed{}.
|
x=0
|
user
|
Balance the following reaction: {eq}Mn + HI \to H_2 + MnI_3
{/eq} Please reason step by step, and put your final answer within \boxed{}.
|
2Mn + 6HI -> 3H_2 + 2MnI_3
|
End of preview. Expand
in Data Studio
No dataset card yet
- Downloads last month
- -
Size of downloaded dataset files:
13.2 MB
Size of the auto-converted Parquet files:
5.98 MB
Number of rows:
34,959