Answer:
a) V = (8 -4/3π)r³
b) 2048(2 -π/3) cm³ ≈ 1951.34 cm³
Step-by-step explanation:
When a sphere is placed in a box the empty space volume is the difference between the box volume and the sphere volume. Here, the box is a cube with the same outside dimensions as the sphere.
__
a. (formula)
The side length of the cube is 2r, where r is the radius of the sphere. So, the volume of the cube is ...
V = s³ = (2r)³ = 8r³
The volume of the sphere is given by the formula ...
V = 4/3πr³
The volume of the empty space is the difference of these:
V = (8r³) -(4/3πr³)
V = (8 -4/3π)r³ . . . . . . . formula for the empty volume
__
b. (empty space)
For r=8 cm, the empty volume is ...
V = (8 -4/3π)(8 cm)³ = 4(2 -π/3)(512 cm³)
V = 2048(2 -π/3) cm³ ≈ 1951.34 cm³ . . . . empty volume
