Question

trash cans a cylindrical trash can is 30 inches high and has a base radius of 7 inches. a manufacturer wants to know the surface area of this trash can, including the top of the lid. what is the surface area? round to the nearest square inch.

in²

Explanation:

Step1: Recall the formula for the surface area of a cylinder

The surface area \( S \) of a cylinder (including the top and bottom) is given by the formula \( S = 2\pi r^2+ 2\pi rh \), where \( r \) is the radius of the base and \( h \) is the height of the cylinder. Here, \( r = 7 \) inches and \( h = 30 \) inches.

Step2: Calculate the area of the two circular bases

The area of one circular base is \( \pi r^2 \), so for two bases, it is \( 2\pi r^2 \). Substituting \( r = 7 \), we get \( 2\times\pi\times7^2=2\times\pi\times49 = 98\pi \).

Step3: Calculate the lateral (curved) surface area

The lateral surface area is \( 2\pi rh \). Substituting \( r = 7 \) and \( h = 30 \), we get \( 2\times\pi\times7\times30=420\pi \).

Step4: Calculate the total surface area

Add the area of the two bases and the lateral surface area: \( S=98\pi + 420\pi=518\pi \).

Step5: Approximate the value

Using \( \pi\approx3.1416 \), we have \( S\approx518\times3.1416 \). Calculating \( 518\times3.1416 = 518\times3+518\times0.1416=1554 + 73.3488 = 1627.3488 \). Rounding to the nearest square inch, we get \( 1627 \).

Answer:

\( 1627 \)