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. 372 ft.
B. 6.5 ft.
C. 57.3 ft.
D. 9.3 ft.
The surface area of a triangular pyramid can be found using the formula:
Surface Area = (base area) + (lateral area)
The base area is given by the formula for the area of a triangle:
Base Area = (1/2)(base length)(height)
We need to find the height in order to calculate the base area. To do this, we can use the Pythagorean theorem for the right triangle formed by the slant height, the height of the pyramid, and half of the base. The equation for the Pythagorean theorem is:
(slant height)² = (height)² + (half of base)²
Plugging in the known values:
8² = (height)² + (3)²
64 = (height)² + 9
(height)² = 64 - 9
(height)² = 55
height ≈ √55
Now that we have the height, we can calculate the base area:
Base Area = (1/2)(6)(√55)
Base Area ≈ 3√55
Finally, we substitute the given surface area and the calculated base area into the surface area formula and solve for the lateral area:
100 = 3√55 + Lateral Area
Lateral Area = 100 - 3√55
Therefore, the height from the base to the apex is equivalent to the lateral height of the surface, which is equal to the slant height of the pyramid. Hence, the height is approximately 8 ft.