Question

Determine the ratio of the volume of a cube to that of the sphere which will exactly fit inside the cube.

Answer 1

Step 1: Cube Volume. Given a cube with side length $a$, its volume is $V_{\text{cube}} = a^3$. Step 2: Sphere Volume. A sphere inscribed within the cube has a diameter equal to the cube's side, $a$. Its radius is $r = \frac{a}{2}$. The sphere's volume is $V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left(\frac{a}{2}\right)^3$. Simplifying, $V_{\text{sphere}} = \frac{4}{3} \pi \cdot \frac{a^3}{8} = \frac{\pi a^3}{6}$. Step 3: Ratio Calculation. The ratio of the cube's volume to the sphere's volume is $\text{Ratio} = \frac{V_{\text{cube}}}{V_{\text{sphere}}} = \frac{a^3}{\frac{\pi a^3}{6}} = \frac{6}{\pi}$. Correct Answer: The ratio is $6 : \pi$.