A music award is a square pyramid with a 6-inch-long base and a 13-inch slant height. Find the award’s total surface area.

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To find the total surface area of the music award, we need to consider the area of the base and the area of the four triangular faces.
First, let's calculate the area of the base. Since the base is a square pyramid with a 6-inch-long base, the area of the base is simply the area of a square, which is the length of one side squared. In this case, the length of one side is 6 inches, so the area of the base is 6^2 = 36 square inches.
Next, let's calculate the area of the triangular faces. Each triangular face is an isosceles triangle, with two equal sides and a base. We are given the slant height of the pyramid, which is 13 inches.
To find the length of the equal sides of the triangle, let's use the Pythagorean theorem. The slant height, the height of the triangle, and one of the equal sides form a right triangle. The height is the perpendicular distance from the apex of the pyramid to the base.
Using the Pythagorean theorem, we can find the length of the equal sides (sides with length h):
h^2 = l^2 - (b/2)^2
h^2 = 13^2 - (6/2)^2
h^2 = 169 - 9
h^2 = 160
h = sqrt(160)
h ≈ 12.65 inches
Now, we can calculate the area of one triangular face using the formula for the area of an isosceles triangle:
Area of a triangle = (1/2) * base * height
Area of a triangle = (1/2) * 6 * 12.65 ≈ 37.95 square inches
Since the pyramid has four triangular faces, the total surface area of the pyramid is:
Total surface area = Base area + 4 * Triangle area
Total surface area = 36 + 4 * 37.95
Total surface area ≈ 36 + 151.8
Total surface area ≈ 187.8 square inches
Therefore, the music award's total surface area is approximately 187.8 square inches.