Doug has a bucket that is 12in tall and has a radius of 6in filled completely with water. Dough pushes a basketball with a diameter of 10in completely into the water overflowing the water in the bucket. How much water is left in the bucket?
First, let's find the volume of water that was in the bucket before the basketball was pushed in.
The volume of a cylinder (the shape of the bucket) is given by the formula V = πr^2h, where r is the radius and h is the height.
Given that the radius is 6 inches and the height is 12 inches, the initial volume of water in the bucket is:
V_initial = π(6)^2(12) = 432π cubic inches
Next, let's find the volume of the basketball. The volume of a sphere (the shape of a basketball) is given by the formula V = 4/3 πr^3, where r is the radius.
Given that the diameter of the basketball is 10 inches (radius = 5 inches), the volume of the basketball is:
V_ball = 4/3 π(5)^3 = 523.6 cubic inches
Since the basketball was pushed completely into the water, it displaced an equal volume of water. Therefore, the final volume of water in the bucket is:
V_final = V_initial - V_ball
V_final = 432π - 523.6
V_final ≈ 108.8 cubic inches
Therefore, there is approximately 108.8 cubic inches of water left in the bucket after the basketball is pushed in.