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2 years ago

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The volume V (in cubic feet) of a right cylinder with a height of 3 feet and radius r (in feet) is given by V=3πr2. Solve the fo

rmula for r. Then find the radius of the cylinder when the volume is 236 cubic feet. Round your answer to the whole number.
The formula solved for r is r=
When the volume is 236 cubic feet, the radius is about __ feet.

1 answer:

makkiz [27]2 years ago

6 0

Answer:

118. The radius is 118 feet.

Step-by-step explanation:

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Use green's theorem to compute the area inside the ellipse x252+y2172=1. use the fact that the area can be written as ∬ddxdy=12∫

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The area of the ellipse $E$ is given by

$\displaystyle\iint_E\mathrm dA=\iint_E\mathrm dx\,\mathrm dy$

To use Green's theorem, which says

$\displaystyle\int_{\partial E}L\,\mathrm dx+M\,\mathrm dy=\iint_E\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)\,\mathrm dx\,\mathrm dy$

( $\partial E$ denotes the boundary of $E$ ), we want to find $M(x,y)$ and $L(x,y)$ such that

$\dfrac{\partial M}{\partial x}-\dfrac{\partial L}{\partial y}=1$

and then we would simply compute the line integral. As the hint suggests, we can pick

$\begin{cases}M(x,y)=\dfrac x2\\\\L(x,y)=-\dfrac y2\end{cases}\implies\begin{cases}\dfrac{\partial M}{\partial x}=\dfrac12\\\\\dfrac{\partial L}{\partial y}=-\dfrac12\end{cases}\implies\dfrac{\partial M}{\partial x}-\dfrac{\partial L}{\partial y}=1$

The line integral is then

$\displaystyle\frac12\int_{\partial E}-y\,\mathrm dx+x\,\mathrm dy$

We parameterize the boundary by

$\begin{cases}x(t)=5\cos t\\y(t)=17\sin t\end{cases}$

with $0\le t\le2\pi$ . Then the integral is

$\displaystyle\frac12\int_0^{2\pi}(-17\sin t(-5\sin t)+5\cos t(17\cos t))\,\mathrm dt$

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###

Notice that $x^{2/3}+y^{2/3}=4^{2/3}$ kind of resembles the equation for a circle with radius 4, $x^2+y^2=4^2$ . We can change coordinates to what you might call "pseudo-polar":

$\begin{cases}x(t)=4\cos^3t\\y(t)=4\sin^3t\end{cases}$

which gives

$x(t)^{2/3}+y(t)^{2/3}=(4\cos^3t)^{2/3}+(4\sin^3t)^{2/3}=4^{2/3}(\cos^2t+\sin^2t)=4^{2/3}$

as needed. Then with $0\le t\le2\pi$ , we compute the area via Green's theorem using the same setup as before:

$\displaystyle\iint_E\mathrm dx\,\mathrm dy=\frac12\int_0^{2\pi}(-4\sin^3t(12\cos^2t(-\sin t))+4\cos^3t(12\sin^2t\cos t))\,\mathrm dt$

$=\displaystyle24\int_0^{2\pi}(\sin^4t\cos^2t+\cos^4t\sin^2t)\,\mathrm dt$

$=\displaystyle24\int_0^{2\pi}\sin^2t\cos^2t\,\mathrm dt$

$=\displaystyle6\int_0^{2\pi}(1-\cos2t)(1+\cos2t)\,\mathrm dt$

$=\displaystyle6\int_0^{2\pi}(1-\cos^22t)\,\mathrm dt$

$=\displaystyle3\int_0^{2\pi}(1-\cos4t)\,\mathrm dt=6\pi$

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2 years ago

What do the differences between the points (as shown on the graph) represent? On a coordinate plane, point B is at (2, 7) and po

raketka [301]

Answer:

StartFraction 7 over 2 EndFraction can be used to find the unit rate if one divides 7 by 2 and compares the result to 1

Step-by-step explanation:

we know that

The formula to calculate the slope between two points is equal to

$m=\frac{y2-y1}{x2-x1}$

we have the points

B(2, 7) and D(4, 14)

substitute the values

$m=\frac{14-7}{4-2}$

$m=\frac{7}{2}$

The unit rate is

$m=\frac{3.5}{1}$

therefore

StartFraction 7 over 2 EndFraction can be used to find the unit rate if one divides 7 by 2 and compares the result to 1

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1 year ago

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A sixth grade teacher takes students on a field trip to the beach. One student finds several pebbles that have a rounded shape a

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Answer:

The shape of the pebbles is a result of weathering and deposition.

Step-by-step explanation:

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2 years ago

The Office of Student Services at UNC would like to estimate the proportion of UNC's 34,000 students who are foreign students. I

Pachacha [2.7K]

Answer:

Mean is 3.05

Standard deviation is 1.69.

Step-by-step explanation:

For each student, there are only two possible outcomes. Either they are foreign, or they are not. This means that we can solve this problem using concepts of the binomial probability distribution.

Binomial probability distribution

The probability of x sucesses on n repeated trials, with p probability.

In this problem, we have a sample of 50 students, so $n = 50$ . The proportion of all UNC students that are foreign students is 0.061, so $p = 0.061$ .

The mean is given by the following formula:

$E(X) = np = 50*0.061 = 3.05$

The standard deviation is given by the following formula:

$\sigma = \sqrt{n*p*(1-p)} = \sqrt{50*0.061*(1-0.061))} = 1.69$

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1 year ago

Is 9.373 a repeating decimal? Is it rational? Explain your reasoning.

Galina-37 [17]

its is not a repeating decimal and it is a rational number because a rational number is a number that does not have repeating decimal but a irrational number does have repeating decimal

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2 years ago

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