Calculate the surface area of a triangular prism shaped tent if the height is 9 ft., the length is 24 ft., the width is 20 ft., and the hypotenuse of the cover is 13.45 ft.
First, we need to calculate the area of each triangular end of the prism. We can do this using the formula for the area of a triangle:
Area of triangle = 0.5 x base x height
The height of each end triangle is 9 ft, and the base is half the width, which is 10 ft. To find the length of the other side of the triangle, we can use the Pythagorean theorem: a^2 + b^2 = c^2. In this case, a and b are the two sides of the triangle that we know (9 ft and 10 ft), and c is the hypotenuse of the cover, which is 13.45 ft. Solving for the missing side, we get:
a^2 + b^2 = c^2
9^2 + 10^2 = 13.45^2
81 + 100 = 181.4
sqrt(181.4) = 13.47
So the length of each side of the triangular end is approximately 13.47 ft. Using the formula above, we can find the area of each end:
Area of end triangle = 0.5 x base x height
= 0.5 x 10 ft x 9 ft
= 45 sq ft
To calculate the area of the rectangular sides of the prism, we can use the formula for the area of a rectangle:
Area of rectangle = length x width
The length of the rectangle is 24 ft and the width is 20 ft, so the area of each of the two rectangular sides is:
Area of rectangular side = length x width
= 24 ft x 20 ft
= 480 sq ft
Adding up the areas of the two end triangles and the two rectangular sides, we get the total surface area of the triangular prism tent:
Total surface area = 2 x area of end triangle + 2 x area of rectangular side
= 2 x 45 sq ft + 2 x 480 sq ft
= 90 sq ft + 960 sq ft
= 1050 sq ft
Therefore, the surface area of the triangular prism tent is 1050 square feet.