Question

Cole has a cylinder that fits within a cube. The height and diameter of the cylinder are each equal to x. Each side of the cube is also equal to x. Cole wants to know the volume of water that can be poured inside the rectangular prism yet outside the cylinder. Which expression will help Cole solve for this volume ?

Answer 1

From the question,

The formula for the volume(V1) of the cube is,

$V_1=l^3$

Given that:

The length of the cube is x

Therefore,

$V_1=x^3$

The formula for the volume (V2) of the cylinder is,

$V_2=\pi r^2h$

Given:

$\begin{gathered} r=\frac{\text{diameter}}{2}=\frac{x}{2} \\ h=\text{height}=x \end{gathered}$

Therefore,

$\begin{gathered} V_2=\pi\times(\frac{x}{2})^2\times x=\pi\times\frac{x^2}{4}\times x=\frac{\pi x^3}{4} \\ \therefore V_2=\frac{\pi x^3}{4} \end{gathered}$

Hence, the volume(V) of water that can be poured inside the rectangular prism yet outside the cylinder will be

$\begin{gathered} V=x^3-\frac{\pi x^3}{4} \\ \therefore V=x^3-\frac{\pi}{4}x^3 \end{gathered}$

Therefore, the expression that will help Cole solve for the volume is,

$x^3-\frac{\pi}{4}x^3$