Final answer:
The surface area of a square pyramid with the given dimensions is calculated by finding the area of the base and the four triangular faces, and then summing those areas, which results in a total surface area of 75 square miles.
Step-by-step explanation:
Surface Area of a Square Pyramid
To find the surface area of a square pyramid with a side length of 5 mi and a slant height of 5 mi, we need to calculate the area of the base and the area of the four triangular faces. The base is a square so its area is the side length squared:
Base area = side length × side length = 5 mi × 5 mi = 25 sq mi.
Each triangular face has a base of 5 mi (the side of the square) and a slant height of 5 mi. The area of one triangle is:
Triangle area = 1/2 × base × slant height = 1/2 × 5 mi × 5 mi = 12.5 sq mi.
Since there are four identical triangles, the total area of the triangular faces is 4 times the area of one triangle:
Total area of triangular faces = 4 × 12.5 sq mi = 50 sq mi.
Finally, the total surface area of the pyramid is the sum of the bases area and the triangles' areas:
Total surface area = base area + total area of triangular faces = 25 sq mi + 50 sq mi = 75 sq mi.
Therefore, the surface area of the pyramid is 75 square miles.