Question

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

Answer 1

Answer:

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

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

Step-by-step 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

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

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

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

$\implies 5=2r$

$\implies r=\dfrac52$

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

$\begin{aligned}\textsf{Volume of a sphere} & =\dfrac43 \pi r^3\\\\ & =\dfrac43 \pi \left(\dfrac52\right)^3\\\\ & =\dfrac43 \pi \left(\dfrac{125}{8}\right)\\\\ & =\dfrac{500}{24} \pi\\\\ & =\dfrac{125}{6} \pi\\\\ & =65.45\:\sf in^3\:(nearest\:hundredth) \end{aligned}$