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: Volume of the cube Let the side of the cube be $a$. The volume of the cube is: $V_{\text{cube}} = a^3.$ Step 2: Volume of the sphere The sphere that fits exactly inside the cube will have a diameter equal to the side of the cube, $a$. The radius of the sphere is: $r = \frac{a}{2}.$ The volume of the sphere is: $V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left(\frac{a}{2}\right)^3.$ Simplify: $V_{\text{sphere}} = \frac{4}{3} \pi \cdot \frac{a^3}{8} = \frac{\pi a^3}{6}.$ Step 3: Find the ratio The ratio of the volume of the cube to the volume of the sphere 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 $6 : \pi$.

Answer 2

Step 1: Volume of the cube Let the side of the cube be $a$. The volume of the cube is: $V_{\text{cube}} = a^3.$ Step 2: Volume of the sphere The sphere that fits exactly inside the cube will have a diameter equal to the side of the cube, $a$. The radius of the sphere is: $r = \frac{a}{2}.$ The volume of the sphere is: $V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left(\frac{a}{2}\right)^3.$ Simplify: $V_{\text{sphere}} = \frac{4}{3} \pi \cdot \frac{a^3}{8} = \frac{\pi a^3}{6}.$ Step 3: Find the ratio The ratio of the volume of the cube to the volume of the sphere 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 $6 : \pi$.