The surface area of the prism can be calculated by adding the area of the two triangular bases and the area of the three rectangular faces.
The area of the triangular base is:
Area = (1/2) * base * height = (1/2) * 10 in * 9 in = 45 in^2
The area of the three rectangular faces is:
Area = length * width
For the rectangular face adjacent to the base triangle, the length is the hypotenuse of the base triangle, which is 13.5 inches, and the width is the height of the prism, which is 9 inches. Therefore, the area is:
Area = 13.5 in * 9 in = 121.5 in^2
For the other two rectangular faces, the length is the hypotenuse of a right triangle with legs of 9 inches and 10 inches, which can be found using the Pythagorean theorem:
c^2 = a^2 + b^2
c^2 = 9^2 + 10^2
c = sqrt(81 + 100) = sqrt(181) ≈ 13.45 in
The width is still 9 inches, so the area of each face is:
Area = 13.45 in * 9 in ≈ 121.05 in^2
Therefore, the total surface area of the prism is:
Surface Area = 2 * Area of Base + 3 * Area of Rectangular Face
Surface Area = 2 * 45 in^2 + 3 * 121.05 in^2 ≈ 549.15 in^2
Rounding to two decimal places, the surface area of the prism is approximately 549.15 in^2.