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?
We know that the surface area of a regular triangular pyramid is given by the formula:
Surface Area = Base Area + Lateral Area
The base area of our tent is the area of a triangle, which is given by:
Base Area = (1/2) * base * height
And the lateral area is given by:
Lateral Area = (1/2) * perimeter * slant height
The perimeter of the base can be given by:
Perimeter = 3 * base
We are given that the surface area of the tent is 100 ft^2, the base is 6 ft, and the slant height is 8 ft. We want to find the height.
Substituting the given values into the formulas above, we have:
100 = (1/2) * 6 * height + (1/2) * 3 * 6 * 8
100 = 3 * height + 72
3 * height = 100 - 72
3 * height = 28
height = 28 / 3
height ≈ 9.333
Therefore, the height of the base is approximately 9.333 ft, to the nearest tenth.