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)

Question

Responses

9.3 ft.
9.3 ft.

57.3 ft.
57.3 ft.

6.5 ft.
6.5 ft.

372 ft.

Answer 1

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.

Answer 2

wrong choose a answer choice that makes the most sense

Answer 3

Based on the given options, the answer that makes the most sense is 9.3 ft.