A cylinder has a 10-inch diameter and is 15 inches tall. 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.

Question

339π in3
375π in3
381π in3
392π in3

Answer 1

To find how much water is in the cylinder after placing the 6-inch-diameter ball inside it, we first need to calculate the volume of the cylinder and then subtract the volume of the ball.

1. Calculate the volume of the cylinder. The formula for the volume of a cylinder is:

   $$V=\pi r^{2}h$$

   where $r$ is the radius and $h$ is the height.

   The diameter of the cylinder is 10 inches, so the radius $r$ is:

   $$r=\frac{10}{2}=5\text{ inches}$$

   The height $h$ of the cylinder is 15 inches. Now, we can calculate the volume:

   $$V_{\text{cylinder}}=\pi (5{)}^2(15)=\pi (25)(15)=375\pi \text{ cubic inches}$$

2. Calculate the volume of the ball. The formula for the volume of a sphere is:

   $$V=\frac{4}{3}\pi r^3$$

   The diameter of the ball is 6 inches, so the radius $r$ is:

   $$r=\frac{6}{2}=3\text{ inches}$$

   Now we can calculate the volume of the ball:

   $$V_{\text{ball}}=\frac{4}{3}\pi (3{)}^3=\frac{4}{3}\pi (27)=36\pi \text{ cubic inches}$$

3. Calculate the volume of water in the cylinder. The volume of water is the volume of the cylinder minus the volume of the ball:

   $$V_{\text{water}}=V_{\text{cylinder}}-V_{\text{ball}}=375\pi -36\pi =339\pi \text{ cubic inches}$$

Thus, the amount of water in the cylinder is:

$$\boxed{{\displaystyle 339\pi }}\text{ cubic inches}$$