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?(1 point)
choose one of the answers
9.3 ft.
9.3 ft.
372 ft.
372 ft.
6.5 ft.
6.5 ft.
57.3 ft.
To find the height of the base, we can use the formula for the surface area of a triangular pyramid:
Surface Area = area of base + (1/2)(perimeter of base)(slant height)
Given that the surface area is 100 ft^2, the base is 6 ft, and the slant height is 8 ft, we can substitute these values into the formula:
100 = (1/2)(6)(8) + area of base
Simplifying:
100 = 24 + area of base
Subtracting 24 from both sides:
76 = area of base
Since the base is a regular triangle, we can find the area of the base by using the formula:
Area of base = (sqrt(3)/4)(side length)^2
Given that the side length is 6 ft, we can substitute this value into the formula:
Area of base = (sqrt(3)/4)(6)^2
Area of base = (sqrt(3)/4)(36)
Area of base = 36(sqrt(3))/4
Area of base = 9(sqrt(3))
Now we can go back to the equation 76 = area of base and substitute the value we just found:
76 = 9(sqrt(3))
We can now solve for sqrt(3):
sqrt(3) = 76 / 9
sqrt(3) = 8.44
Finally, we can find the height of the base by using the formula for a regular triangular pyramid:
Height of base = (2 * Surface Area) / (perimeter of base)
Height of base = (2 * 100) / 18
Height of base = 200 / 18
Height of base = 11.1
Rounded to the nearest tenth, the height of the base is 11.1 ft. None of the provided answer choices match this value.