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?
The surface area of a regular triangular pyramid is given by the formula:
Surface area = (base area) + 0.5 * (perimeter of base) * (slant height)
In this case, we are given the surface area as 100 ft^2, the base as 6 ft, and the slant height as 8 ft.
Let's calculate the base area:
base area = (base length * base length * √3) / 4
base area = (6 * 6 * √3) / 4
base area ≈ 15.5885 ft^2
Now, let's use the formula for surface area to find the height:
100 = 15.5885 + 0.5 * (6 + 6 + 6) * (8)
100 = 15.5885 + 0.5 * 18 * 8
100 = 15.5885 + 72
84.4115 = 72
Let h be the height of the pyramid. From the formula above, we can write the equation:
100 = (6 * 6 * √3) / 4 + 0.5 * (6 + 6 + 6) * h
100 = 15.5885 + 18h
18h = 100 - 15.5885
18h = 84.4115
h ≈ 4.6895
The height of the base to the nearest tenth is 4.7 ft. Answer: \boxed{4.7}.