A ball of radius 12 has a round hole of radius 6 drilled through its center. Find the volume of the resulting solid.
To find the volume of the resulting solid, we need to subtract the volume of the hole from the volume of the ball.
The volume of a sphere can be calculated using the formula V = (4/3)πr³, where V is the volume and r is the radius.
So, the volume of the ball with radius 12 is V_ball = (4/3)π(12)³ = 2304π cubic units.
The volume of a cylinder can be calculated using the formula V = πr²h, where V is the volume, r is the radius of the base, and h is the height.
Since the hole is drilled through the center of the ball, it can be visualized as a cylinder with height equal to the diameter of the ball (since the diameter is equal to twice the radius). Therefore, the height of the cylinder is 2r = 2(6) = 12.
The volume of the hole is V_hole = π(6)²(12) = 432π cubic units.
Finally, to find the volume of the resulting solid, we subtract the volume of the hole from the volume of the ball:
V_resulting_solid = V_ball - V_hole = 2304π - 432π = 1872π cubic units.
So, the volume of the resulting solid is 1872π cubic units.