a A film can that represents the results of a survey can be modeled by the following cylinder with a diameter of 12 inches and a radius of r= 122=6 inches.
We are asked to explain how to find the surface area of the portion that represents people who prefer to watch movies at home. Let's analyze this part.
Notice that the surface area of this part is equal to the 73 % of the surface area of full cylinder, plus the area of two lateral rectangles that occurred from the removal of the other parts from the cylinder.
b We are asked to find the surface area of the portion in Part A if we assume that the film can is 3 inches tall.
From Part A, the surface area of the solid is equal to 73 % of the surface area of the full cylinder \textcolor{darkorange}{S_\text{cylinder}}, plus the area of the two lateral rectangles that are 6 by 3 inches. The area of each rectangle is \textcolor{darkviolet}{A_\boxed{\ }}=6\cdot 3=\textcolor{darkviolet}{18} square inches. Now, let's use the formula for the surface area of a cylinder.
Therefore, the surface area of the cylinder is about 339.29 square inches. Now, let's find the surface area of the portion of the cylinder S_\text{portion}.
