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
6.5 ft.
6.5 ft.
372 ft.
372 ft.
9.3 ft.
9.3 ft.
57.3 ft.
To find the height of the triangular pyramid, we first need to find the lateral area of the pyramid. The lateral area is the sum of the areas of the four triangles.
The formula for the lateral area of a triangular pyramid is: LA = (1/2) * base * slant height
Given that the base length is 6 ft and the slant height is 8 ft, we can substitute these values into the formula to find the lateral area.
LA = (1/2) * 6 ft * 8 ft
LA = 24 ft^2
We are also given that the total surface area of the tent is 100 ft^2. This includes the lateral area and the area of the base.
The formula for the total surface area of a triangular pyramid is: TSA = LA + base area
Given that the base length is 6 ft, we can calculate the area of the base using the formula for the area of a triangle: base area = (1/2) * base * height
We can rearrange the formula to solve for the height: height = (2 * base area) / base
base area = (1/2) * 6 ft * height
base area = 3 ft * height
Substituting the values into the formula for the total surface area:
100 ft^2 = 24 ft^2 + 3 ft * height
Simplifying the equation:
76 ft^2 = 3 ft * height
Divide both sides of the equation by 3 ft:
25.33 ft = height
Rounded to the nearest tenth, the height of the base is 25.3 ft.
Therefore, the correct answer is not given in the responses.