Question

A container in the shape of a right circular cylinder is $\frac{1}{2}$ full of water. If the volume of water in the container is $36$ cubic inches and the height of the container is $9$ inches, what is the diameter of the base of the cylinder, in inches?

• A. $\frac{16}{9\pi }$
• B. $\frac{4}{\sqrt{\pi }}$
• C. $\frac{12}{\sqrt{pi}}$
• D. $\sqrt{\frac{2}{\pi }}$
• E. $4\sqrt{\frac{2}{\pi }}$

Answer 1

The correct option is E $4\sqrt{\frac{2}{\pi }}$

For a right cylinder, $volume=\pi \left(radius{)}^2\right(height)$. Since the volume of water is $36$ cubic inches and since this represents $\frac{1}{2}$ the container, the is occupying $\frac{1}{2}$ the container's height, or $9\left(\frac{1}{2}\right)=4.5inches$. Let $r$ be the radius of the cylinder.

$36=\pi r^2\left(4.5\right)$

$8=\pi r^2$ divide both sides by $4.5$

$\frac{8}{\pi }=r^2$ divide both sides by $\pi$

$\sqrt{\frac{8}{\pi }}=r$ take the square root of both sides

$\frac{2\sqrt{2}}{\sqrt{\pi }}=r$ simplify the $\sqrt{8}$ to get the radius

Then, since the diameter is twice the length of the radius, the diameter equals

$2\left(\frac{2\sqrt{2}}{\sqrt{\pi }}\right)=4\frac{\sqrt{2}}{\sqrt{pi}}=4\sqrt{\frac{2}{\pi }}$.

The correct answer is E.