Question

A cylinder with radius r and height 2r+4 contains a cube with edge length r + 2, as shown.

Answer 1

Volume is the amount of space in an object. The fraction occupied by the cube is: $\frac{r\sqrt2}{\pi(r + 2)}$

First, we calculate the volume of the cube.

$Volume = Length^3$

From the figure, we have:

$Length = r\sqrt 2$

So:

$V_1 = (r\sqrt2)^3$

$V_1 = (2\sqrt2)r^3$

Next, the volume of the cylinder

$Volume=\pi r^2h$

So, we have:

$V_2 = \pi r^2 (2r + 4)$

The fraction (n) occupied by the cube is:

$n = \frac{V_1}{V_2}$

So, we have:

$n = \frac{(2\sqrt2)r^3}{\pi r^2(2r + 4)}$

Factor out 2 in the denominator

$n = \frac{(2\sqrt2)r^3}{2\pi r^2(r + 2)}$

Cancel out common terms

$n = \frac{r\sqrt2}{\pi(r + 2)}$

Hence, the fraction occupied by the cube is: $\frac{r\sqrt2}{\pi(r + 2)}$

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