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

User Scagood
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3 votes

Answer:

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}

A sphere is inscribed in a cube with a volume of 125 cubic inches what is the volume-example-1
User Jeremy Coenen
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