A ball of radius 14 has a round hole of radius 8 drilled through its center. Find the volume of the resulting solid
To find the volume of the resulting solid, we first need to determine the volume of the ball and then subtract the volume of the drilled hole.
1. Volume of the ball: The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius. In this case, the ball has a radius of 14, so we can substitute it into the formula:
V_ball = (4/3)π(14)³
= (4/3)π(2744)
≈ 14366.24 cubic units
2. Volume of the drilled hole: The drilled hole is in the shape of a cylinder with a circular base. The formula for the volume of a cylinder is V = πr²h, where r is the radius of the base and h is the height. In this case, the radius of the hole is 8, and the height can be determined as twice the radius of the ball, which is 2 * 14 = 28 units.
V_hole = π(8)² * 28
= π(64) * 28
≈ 5647.91 cubic units
3. Subtract the volume of the drilled hole from the volume of the ball:
V_resulting_solid = V_ball - V_hole
= 14366.24 - 5647.91
≈ 8718.33 cubic units
Therefore, the volume of the resulting solid is approximately 8718.33 cubic units.