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
57.3 ft.
57.3 ft.
9.3 ft.
9.3 ft.
372 ft.
372 ft.
6.5 ft.
To find the height of the tent, we need to use the formula for the surface area of a triangular pyramid, which is given by:
Surface Area = Base Area + Lateral Area
In this case, the base area of the pyramid is a triangle, which can be found using the formula:
Base Area = (1/2) * base * height
We are given that the base is 6 ft, so we need to find the height of the base. Let's denote it as 'h'.
We are also given that the slant height of the pyramid is 8 ft.
Using the Pythagorean theorem, we can find the height of the triangular face of the pyramid:
8^2 = (6/2)^2 + h^2
64 = 9 + h^2
h^2 = 64 - 9
h^2 = 55
h = √55
Now, we can substitute this value for the height of the base into the formula for the base area:
Base Area = (1/2) * 6 * √55
Base Area = 3 * √55
We are also given that the surface area of the tent is 100 ft^2, so we can set up the equation:
100 = 3 * √55 + Lateral Area
We need to solve for the Lateral Area. In a regular triangular pyramid, the lateral area is given by:
Lateral Area = (1/2) * perimeter * slant height
The perimeter of the triangle forming the base can be found by multiplying the base length by 3. So:
perimeter = 3 * 6 = 18
Substituting this value into the equation for Lateral Area:
Lateral Area = (1/2) * 18 * 8
Lateral Area = 72
Now we can substitute this value for the Lateral Area back into the equation for the surface area:
100 = 3 * √55 + 72
Subtracting 72 from both sides:
28 = 3 * √55
Dividing both sides by 3:
√55 = 28/3
Simplifying the right side:
√55 ≈ 9.3
Therefore, the height of the base is approximately 9.3 ft