a. A cube's volume is 512 cubic units. What is the length of its edge? b. If a sphere fits snugly inside this cube, what is its volume?

To find the edge length of the cube, you can use the volume formula: \( V = a^3 \), where \( a \) is the length of the edge. Given the volume is 512 cubic units, you take the cube root: \( a = \sqrt[3]{512} = 8 \) units. So, the edge length of the cube is 8 units! When it comes to the snug-fitting sphere within the cube, its diameter equals the cube's edge length. Thus, the radius \( r \) of the sphere is half of 8 units, making it 4 units. The volume of a sphere is given by the formula \( V = \frac{4}{3} \pi r^3 \). Plugging in the radius: \( V = \frac{4}{3} \pi (4^3) = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \approx 268.08 \) cubic units. So, the volume of the sphere is about 268.08 cubic units!