# Question 53d0e

Jan 2, 2018

Cylinder with radius of $4.3$ $c m$ and a height of $8.6$ $c m$.

#### Explanation:

First, let's recognize that we must find the dimensions of a cylinder that yields a volume of $500$ $c {m}^{3}$ and has the least surface area possible. This means that we must minimize the quantity $2 \pi {r}^{2}$ + $2 \pi r h$.

Since the volume of a cylinder is $\pi {r}^{2} h$, we have

$\pi {r}^{2} h$ = $500$.

$h$ = $\frac{500}{\pi {r}^{2}}$.

Plugging back into our equation for surface area yields

$A \left(r\right)$ = $2 \pi {r}^{2}$ + $\frac{1000}{r}$.

Since we want to find the minimum of this function, we must take the derivative with respect to $r$ and set the derivative equal to 0.

$0$ = $4 \pi r$ + $- \frac{1000}{r} ^ 2$.

Now, all we have to do to solve for $r$. Rearranging yields

$\frac{1000}{r} ^ 2$ = $4 \pi r$

$1000$ = $4 \pi {r}^{3}$

${r}^{3}$ = $\frac{250}{\pi}$

$r = \sqrt[3]{\frac{250}{\pi}}$.

Solving for $h$ with this value of $r$ yields

h = 500/(piroot(3)(250/pi)^2#

$h = \frac{500}{\pi \left({250}^{\frac{2}{3}} / {\pi}^{\frac{2}{3}}\right)}$

$h = \frac{2 \times {250}^{\frac{3}{3}}}{{250}^{\frac{2}{3}} \times \left({\pi}^{\frac{3}{3}} / {\pi}^{\frac{2}{3}}\right)}$

$h = 2 \times {250}^{\frac{1}{3}} / {\pi}^{\frac{1}{3}}$

$h = 2 \sqrt[3]{\frac{250}{\pi}}$.

So our final answer is a cylinder with a radius of $4.3$ $c m$ and height of $8.6$ $c m$.