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)
The surface area of the tent can be calculated by the formula:
Area = Base Area + Lateral Area
The base area of the triangular pyramid is a triangle, and its formula is:
Base Area = 1/2 * Base * Height
Given that the base is 6 ft., we can substitute this value into the formula:
Base Area = 1/2 * 6 ft. * Height
The lateral area of the triangular pyramid is given by the formula:
Lateral Area = 1/2 * Perimeter * Slant Height
The perimeter of the triangular pyramid can be calculated as:
Perimeter = 3 * Base
We know that the base is 6 ft., so we can substitute this value into the formula:
Perimeter = 3 * 6 ft. = 18 ft.
Substituting the values into the formula for the lateral area, we get:
Lateral Area = 1/2 * 18 ft. * 8 ft. = 72 ft.²
Now we can substitute the values into the formula for the surface area:
Area = Base Area + Lateral Area
100 ft.² = 1/2 * 6 ft.* Height + 72 ft.²
100 ft.² - 72 ft.² = 1/2 * 6 ft. * Height
28 ft.² = 3 ft. * Height
Height = 28 ft.² / 3 ft.
Height = 9.33 ft.
Therefore, the height of the base is approximately 9.33 feet.