Question

a cylinder has an 18 - inch diameter and is 15 inches tall. it is filled to the top with water. a 6 - inch - diameter ball is placed within the cylinder, and then the cylinder is filled with water. how much water is in the cylinder? give your answer in terms of pi.
1.162π in³
1.179π in³
1.215π in³
1.251π in³

Explanation:

Step1: Find the radius of the cylinder

The diameter of the cylinder is 18 inches, so the radius $r_{cylinder}=\frac{18}{2}=9$ inches. The height of the water - filled part of the cylinder is 15 inches.

Step2: Find the volume of the cylinder filled with water

The volume formula for a cylinder is $V = \pi r^{2}h$. Substituting $r = 9$ inches and $h = 15$ inches, we get $V_{cylinder}=\pi\times9^{2}\times15=\pi\times81\times15 = 1215\pi$ cubic inches.

Step3: Find the radius of the ball

The diameter of the ball is 6 inches, so the radius $r_{ball}=\frac{6}{2}=3$ inches.

Step4: Find the volume of the ball

The volume formula for a sphere (ball) is $V=\frac{4}{3}\pi r^{3}$. Substituting $r = 3$ inches, we get $V_{ball}=\frac{4}{3}\pi\times3^{3}=\frac{4}{3}\pi\times27 = 36\pi$ cubic inches.

Step5: Find the volume of water in the cylinder

The volume of water in the cylinder is the volume of the water - filled part of the cylinder minus the volume of the ball. So $V_{water}=V_{cylinder}-V_{ball}=1215\pi - 36\pi=1179\pi$ cubic inches.

Answer:

$1,179\pi$ in$^{3}$