16. The net of a right pyramid with a square base is shown. The surface area is 90 square inches. The side of the square base is 6 inches. What is the area, in square inches, of each triangular face?
Answer: A
The area of each triangular face is 13.5 square inches.
To determine the area of each triangular face of the pyramid, we first need to calculate the total lateral surface area of the pyramid and then divide it by the number of triangular faces, which is four. Given the total surface area of 90 square inches and the area of the square base (6 inches x 6 inches = 36 square inches), the lateral surface area can be calculated as 90 - 36 = 54 square inches. Dividing this by 4 gives us an area of 13.5 square inches for each triangular face.
A) 13.5
This option is correct. As calculated, the total lateral surface area is 54 square inches, and when divided among the four triangular faces, each face has an area of 13.5 square inches.
B) 18
This option is incorrect. If each triangular face had an area of 18 square inches, the total lateral surface area would be 18 x 4 = 72 square inches. Adding this to the area of the base would result in a total surface area of 108 square inches, which exceeds the given total surface area of 90 square inches.
C) 22.5
This option is incorrect. An area of 22.5 square inches for each triangular face would lead to a total lateral surface area of 22.5 x 4 = 90 square inches. This would imply that there is no area left for the square base, which is not feasible as the base area must be included in the total surface area.
D) 54
This option is incorrect. If each triangular face had an area of 54 square inches, the total lateral surface area would be 54 x 4 = 216 square inches. This value, when added to the base area, would yield a total surface area of 252 square inches, which is far greater than the provided total surface area of 90 square inches.
Conclusion
The area of each triangular face is definitively 13.5 square inches, as it is derived from accurately calculating the lateral surface area based on the total surface area and the area of the base. All other options fail to satisfy the conditions of the problem, leading to total surface areas that either exceed or do not account for the given dimensions.