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)
Responses
9.3 ft.
9.3 ft.
57.3 ft.
57.3 ft.
372 ft.
372 ft.
6.5 ft.
To solve for the height, we first need to find the lateral area of the triangular pyramid.
The formula for the lateral area of a triangular pyramid is given by:
Lateral Area = (1/2) * perimeter of base * slant height
The base of the pyramid is a triangle with side lengths of 6 ft each. Therefore, the perimeter of the base is 3 * 6 = 18 ft.
Plugging in the values, we have:
Lateral Area = (1/2) * 18 ft * 8 ft = 72 ft^2
Since the surface area of the tent is given as 100 ft^2 and the lateral area is 72 ft, we can find the area of the top of the tent by subtracting the lateral area from the total surface area:
Area of top = Surface Area - Lateral Area = 100 ft^2 - 72 ft^2 = 28 ft^2
The area of the top of the tent is equal to the area of an equilateral triangle with side length equal to the base. The formula for the area of an equilateral triangle is given by:
Area of equilateral triangle = (sqrt(3)/4) * s^2, where s is the side length.
Plugging in the values, we have:
28 ft^2 = (sqrt(3)/4) * 6 ft^2
Simplifying, we get:
sqrt(3)/4 * 6 ft^2 = 28 ft^2
Multiply both sides by 4/sqrt(3):
6 ft^2 = 28 ft^2 * 4/sqrt(3)
Divide both sides by 28 ft^2:
6 ft^2 / 28 ft^2 = 4/sqrt(3)
Simplify:
1/4.6667 = 4/sqrt(3)
Cross-multiply:
1 * sqrt(3) = 4 * 4.6667
Simplify:
sqrt(3) = 18.6668
Therefore, the height of the base to the nearest tenth is 6.5 ft.