Final answer:
The volume of the ring-shaped solid remaining after drilling a hole through a sphere is calculated by finding the volume of the sphere and subtracting the volume of the cylindrical hole.
Step-by-step explanation:
To find the volume of the ring-shaped solid that remains after drilling a hole through a sphere, we need to subtract the volume of the cylinder (the hole) from the volume of the entire sphere. The volume of a sphere can be calculated using the formula V = (4/3)πr^3. For a sphere with a radius of 7 cm, the volume V_s is:
V_s = (4/3)π(7 cm)^3
The volume of a cylindrical hole drilled through the sphere can be calculated using the formula V_c = πr^2h. However, note that the height of the cylinder will be the diameter of the sphere, so the height h is 2 times the radius of the sphere (2 × 7 cm = 14 cm). Given that the radius of the drill (cylinder) is 3 cm, the volume V_c of the cylinder is:
V_c = π(3 cm)^2(14 cm)
Therefore, the volume of the ring-shaped solid V_ring is:
V_ring = V_s - V_c
After calculating both volumes, plug in the values and subtract V_c from V_s to find the volume of the ring-shaped solid remaining.