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.
6.5 ft.
6.5 ft.
372 ft.
To find the height, we need to use the formula for the surface area of a regular triangular pyramid:
Surface Area = (base area) + (lateral area)
Since the base is a regular triangle, we can find the base area using the formula:
Base Area = (1/2) * base * height
We can use the Pythagorean theorem to find the height of the base triangle:
height^2 + (base/2)^2 = slant height^2
Plugging in the values given:
height^2 + (6/2)^2 = 8^2
height^2 + 3^2 = 8^2
height^2 + 9 = 64
height^2 = 55
Taking the square root of both sides:
height = √55 ≈ 7.4 ft
Now we can find the base area:
Base Area = (1/2) * 6 * 7.4
Base Area ≈ 22.2 ft²
Finally, we can plug the values into the surface area formula:
100 = 22.2 + (lateral area)
To find the lateral area, we subtract the base area from the surface area:
lateral area = 100 - 22.2
lateral area ≈ 77.8 ft²
The lateral area of a regular triangular pyramid is given by the formula:
lateral area = (perimeter of base) * (slant height) / 2
Since the base is a regular triangle, the perimeter is equal to 3 times the length of one side:
lateral area = 3 * side length * slant height / 2
We can solve for the side length:
77.8 = 3 * side length * 8 / 2
77.8 = 12 * side length
side length = 77.8 / 12
side length ≈ 6.5 ft
The height is the height of the base triangle, which we found to be approximately 7.4 ft.