Question

A sphere is inscribed in a cube with a volume of 125 cubic inches what is the volume of the sphere

Answer 1

using
`\pi`: 65.45 in³ (nearest hundredth)

using
`\pi =3.14`: 65.42 in³ (nearest hundredth)

Explanation:

The radius of the sphere is half the side length of the cube (see attached diagram). Therefore, the side length of the cube = 2r

Given:

• volume of the cube = 125 in³
• side length of cube = 2r

`\textsf{Volume of a cube}=x^3\quad \textsf{(where}\:x\:\textsf{is the side length)}`

`\implies 125=(2r)^3`

`\implies \sqrt[3]{125}=2r`

`\implies 5=2r`

`\implies r=\frac52`

Substitute the found value of r into the volume of a sphere equation:

`\begin{aligned}\textsf{Volume of a sphere} & =\frac43 \pi r^3\\\\ & =\frac43 \pi \left(\frac52\right)^3\\\\ & =\frac43 \pi \left((125)/(8)\right)\\\\ & =(500)/(24) \pi\\\\ & =(125)/(6) \pi\\\\ & =65.45\:\sf in^3\:(nearest\:hundredth) \end{aligned}`