Final answer:
To calculate the free space left in the box, subtract the volume of the sphere from the volume of the cube.
Step-by-step explanation:
To find the amount of free space left in the box, we need to calculate the volume of the cube and the volume of the sphere. The volume of a cube with side length (a) is given by V = a^3. Since the diameter of the sphere is equal to the side length of the cube, the radius of the sphere is half the side length of the cube, or r = (a/2). The volume of the sphere is given by V = (4/3)πr^3. To calculate the free space, subtract the volume of the sphere from the volume of the cube:
Free Space = Cube Volume - Sphere Volume
Free Space = a^3 - (4/3)πr^3