Question

a cube is inscribed in a sphere. find the ratio of the volume of the sphere to the volume of the cube.
the ratio of the volume of the sphere to the volume of the cube is
(simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Let the side - length of the cube be \(a\).

The volume of the cube \(V_{cube}=a^{3}\).

Step2: Find the relationship between the side - length of the cube and the radius of the sphere.

When a cube is inscribed in a sphere, the diameter \(d\) of the sphere is equal to the space - diagonal of the cube. The space - diagonal of a cube with side - length \(a\) is \(d = \sqrt{a^{2}+a^{2}+a^{2}}=\sqrt{3}a\), so the radius of the sphere \(r=\frac{\sqrt{3}a}{2}\).

Step3: Calculate the volume of the sphere.

The volume of a sphere is \(V_{sphere}=\frac{4}{3}\pi r^{3}\). Substitute \(r = \frac{\sqrt{3}a}{2}\) into the formula: \(V_{sphere}=\frac{4}{3}\pi(\frac{\sqrt{3}a}{2})^{3}=\frac{4}{3}\pi\times\frac{3\sqrt{3}a^{3}}{8}=\frac{\sqrt{3}\pi a^{3}}{2}\).

Step4: Calculate the ratio of the volume of the sphere to the volume of the cube.

The ratio \(\frac{V_{sphere}}{V_{cube}}=\frac{\frac{\sqrt{3}\pi a^{3}}{2}}{a^{3}}=\frac{\sqrt{3}\pi}{2}\).

Answer:

\(\frac{\sqrt{3}\pi}{2}\)