Question

an air duct in a stadium has a circular cross section with a radius of 15 inches and a length of 35 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 171.81 ft³. (round to the nearest hundredth as needed.) painting the duct requires □ of paint. (round to the nearest hundredth as needed.)

Explanation:

Step1: Convert radius to feet

The radius is 15 inches. Since 1 foot = 12 inches, we convert inches to feet: $r=\frac{15}{12} = 1.25$ feet.

Step2: Recall the formula for the lateral surface area of a cylinder

The duct is a cylinder open at both ends, so we need the lateral (curved) surface area. The formula for the lateral surface area of a cylinder is $SA = 2\pi rh$, where $r$ is the radius and $h$ is the height (length) of the cylinder.

Step3: Substitute the values into the formula

We know $r = 1.25$ feet and $h = 35$ feet. Plugging these into the formula: $SA=2\times\pi\times1.25\times35$.
First, calculate $2\times1.25\times35 = 2.5\times35 = 87.5$. Then multiply by $\pi$: $SA = 87.5\pi\approx87.5\times3.1416\approx274.89$ square feet.

Answer:

274.89