Questions: An air duct in a stadium has a circular cross section with a radius of 12 inches and a length of 36 feet and is open at both ends. What is the volume of the duct, and how much paint (in square feet) is needed to paint the exterior of the duct? The volume of the duct is (Round to the nearest hundredth as needed.) Painting the duct requires of paint. (Round to the nearest hundredth as needed.)

An air duct in a stadium has a circular cross section with a radius of 12 inches and a length of 36 feet and is open at both ends. What is the volume of the duct, and how much paint (in square feet) is needed to paint the exterior of the duct?

The volume of the duct is (Round to the nearest hundredth as needed.) Painting the duct requires of paint. (Round to the nearest hundredth as needed.)

Transcript text: An air duct in a stadium has a circular cross section with a radius of 12 inches and a length of 36 feet and is open at both ends. What is the volume of the duct, and how much paint (in square feet) is needed to paint the exterior of the duct? The volume of the duct is $\square$ $\square$ (Round to the nearest hundredth as needed.) Painting the duct requires $\square$ $\square$ of paint. (Round to the nearest hundredth as needed.)

Solution

Solution Steps

Step 1: Convert all measurements to a consistent unit system

To ensure consistency in units, the radius is converted from inches to feet. Given that 1 inch = 1/12 feet, a radius of 12 inches is equivalent to 1 feet.

Step 2: Calculate the volume of the duct

Using the formula for the volume of a cylinder, $V = \pi r^2 L$, where $r$ is 1 feet and $L$ is 36 feet, the volume is calculated as 113.1 cubic feet.

Step 3: Calculate the surface area needed to paint

Using the formula for the lateral surface area of a cylinder, $A = 2\pi r L$, where $r$ is 1 feet and $L$ is 36 feet, the surface area needed to paint is calculated as 226.19 square feet.