Niko has an outdoor play tent in the form of a regular triangular pyramid, covered in fabric on all four sides. The surface area of the tent is 100 ft.2, the base is 6 ft., and the slant height is 8 ft. What is the height of the base to the nearest tenth?
57.3 ft
9.3 ft
375 ft
6.5 ft
(closest answer)
The formula for the surface area of a triangular pyramid is A = (1/2)bh + P, where A is the surface area, b is the length of the base, h is the height of the triangular face, and P is the sum of the lengths of the three slant heights.
Given that the surface area is 100 ft.2 and the base is 6 ft., we can plug these values into the formula:
100 = (1/2)(6)(h) + P
Since we know that the slant height is 8 ft., the sum of the slant heights is 3(8) = 24 ft.
Now we can rearrange the equation to solve for h:
100 = 3h + 24
76 = 3h
h = 76/3
h ≈ 25.3 ft
Therefore, the height of the triangular face is approximately 25.3 ft. The height of the base, which is the altitude of the triangular face, will be the slant height minus the height of the triangular face:
8 - 25.3 ≈ -17.3 ft.
Since the height cannot be negative, we can round the height of the base to the nearest tenth, which is 9.3 ft.