5. The caps on the tops of staircase posts are shaped like square pyramids. The side length of the base of each cap is 4 inches. The height of the face of each cap is 5 inches. What is the surface area of the caps for two posts?

To find the surface area of the pyramid-shaped caps for two posts, we first calculate the surface area of one cap. The base is a square with side length 4 inches, which gives the area of the base as \(A_{base} = 4 \times 4 = 16\) square inches. Next, we calculate the area of the four triangular faces. The slant height can be found using the Pythagorean theorem, considering half the base length (2 inches) and the height of the triangular face (5 inches): \[ l = \sqrt{(2^2 + 5^2)} = \sqrt{4 + 25} = \sqrt{29} \text{ inches} \] The area of one triangular face is \(\frac{1}{2} \times base \times height\), which is \(\frac{1}{2} \times 4 \times 5 = 10\) square inches. Thus, the total area for the four triangular faces is \(4 \times 10 = 40\) square inches. Adding the base area to the area of the triangular faces gives: \[ S = A_{base} + A_{triangular faces} = 16 + 40 = 56 \text{ square inches} \] Since there are two posts, the combined surface area is: \[ S_{total} = 2 \times S = 2 \times 56 = 112 \text{ square inches} \] So, the surface area of the caps for two posts is 112 square inches.