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?
A.6.5 ft.
B.57.3 ft.
C.372 ft.
D.9.3 ft.
The surface area of a regular triangular pyramid can be found by the formula:
SA = (1/2)(p)(s) + (b)(h)
where SA is the surface area, p is the perimeter of the base, s is the slant height, b is the base area, and h is the height of the pyramid.
In this case, we are given that the surface area is 100 ft^2, the base is 6 ft, and the slant height is 8 ft.
We can substitute the given values into the formula:
100 = (1/2)(6)(8) + (6)(h)
Simplifying the equation:
100 = 24 + 6h
Subtracting 24 from both sides:
76 = 6h
Dividing both sides by 6:
h = 12.67 ft
Rounding to the nearest tenth:
h = 12.7 ft
The height of the base of the tent is approximately 12.7 ft.
Therefore, the answer is option D. 9.3 ft (which is incorrect).