Problem Solving

A cylinder is placed inside a cube so that it stands upright when the cube rests on one of its faces. If the volume of the cube is 16, what is the maximum possible volume of the cylinder that fits inside the cube as described?

A. 16/Ï€
B. 2Ï€
C. 8
D. 4Ï€
E. 8Ï€

Hi lheiannie07,
Let's take a look at your question.

Volume of the cube = 16
If x represents the side length of the cube then,
$$x^3=16$$
$$x=\sqrt[3]{16}$$

Since, the cylinder is placed inside the cube, therefore its maximum radius and height will be equal to the side length of the cube.
Hence,
$$Height=h=\sqrt[3]{16}$$
$$Diameter=\sqrt[3]{16}$$
$$Radius=r=\frac{\sqrt[3]{16}}{2}$$

Volume of the cylinder canbe calculated using formula:
$$=\pi r^2h$$
$$=\pi\left(\frac{\sqrt[3]{16}}{2}\right)^{^{^2}}\left(\sqrt[3]{16}\right)$$
$$=\pi\left(\frac{16^{\frac{2}{3}}}{4}\right)\left(16^{\frac{1}{3}}\right)$$
$$=\frac{\pi}{4}.\left(16^{\frac{2}{3}+\frac{1}{3}}\right)$$
$$=\frac{\pi}{4}.\left(16^{\frac{3}{3}}\right)$$
$$=\frac{\pi}{4}.\left(16\right)$$
$$=4\pi$$

Therefore, Option D is correct.

Hope it helps.
I am available if you'd like any follow up.

lheiannie07 wrote:A cylinder is placed inside a cube so that it stands upright when the cube rests on one of its faces. If the volume of the cube is 16, what is the maximum possible volume of the cylinder that fits inside the cube as described?

A. 16/Ï€
B. 2Ï€
C. 8
D. 4Ï€
E. 8Ï€

Can some experts help me find the best solution in this problem?

OA D

If 2a is the side of the cube its volume would be 8a^3=16
a^3= 2
Diameter and height of the biggest cylinder that fits in cube would be 2a
and its volume would be =π(a)^2×2a=2πa^3 =4 π............( because a^3=2)
hence option D is correct.
.