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 (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: 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 (length) of the cylinder. Here, $r = 1.25$ feet and $h=35$ feet.
Substitute the values into the formula: $V=\pi\times(1.25)^{2}\times35$.
First, calculate $(1.25)^{2}=1.5625$. Then, $1.5625\times35 = 54.6875$. Then, $V=\pi\times54.6875\approx3.14159\times54.6875\approx171.77$ cubic feet.

Step3: Calculate the lateral surface area (for paint)

The formula for the lateral surface area of a cylinder (since it's open at both ends, we don't paint the circular ends) is $A = 2\pi rh$.
Substitute $r = 1.25$ feet and $h = 35$ feet: $A=2\times\pi\times1.25\times35$.
First, $2\times1.25\times35=87.5$. Then, $A = \pi\times87.5\approx3.14159\times87.5\approx274.89$ square feet.

Answer:

(Volume):
$\approx 171.77$ cubic feet