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 volume of 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. Plugging in the values: $V = \pi \times (1.25)^2 \times 35$.
First, calculate $(1.25)^2 = 1.5625$. Then, $1.5625 \times 35 = 54.6875$. Then, $V = \pi \times 54.6875 \approx 3.14159 \times 54.6875 \approx 171.77$ cubic feet.

Step3: Calculate surface area for painting (lateral surface area)

The formula for the lateral surface area of a cylinder (since it's open at both ends, we don't include the top and bottom areas) is $A = 2\pi r h$. Plugging in $r = 1.25$ feet and $h = 35$ feet: $A = 2 \times \pi \times 1.25 \times 35$.
First, calculate $2 \times 1.25 = 2.5$. Then, $2.5 \times 35 = 87.5$. Then, $A = \pi \times 87.5 \approx 3.14159 \times 87.5 \approx 274.89$ square feet.

Answer:

The volume of the duct is approximately $\boldsymbol{171.77}$ cubic feet, and the amount of paint needed (lateral surface area) is approximately $\boldsymbol{274.89}$ square feet. (For the volume part as asked in the box, the answer is approximately 171.77 cubic feet)