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 100ft.^2 , the base is 6 ft., and the slanted height is 8 ft. What is the height of the base to the nearest tenth? (1 point)
372 ft.
57.3 ft.
9.3 ft.
6.5 ft.
The surface area of a triangular pyramid is given by the formula:
Surface area = base area + lateral area
The base area of a regular triangular pyramid is given by the formula:
Base area = (1/2) * base * base to height ratio
The lateral area of a regular triangular pyramid is given by the formula:
Lateral area = (1/2) * perimeter * slant height
First, let's calculate the base area:
Base area = (1/2) * 6 ft * (base to height ratio)
Next, let's calculate the lateral area:
Lateral area = (1/2) * (perimeter) * 8 ft
Since we know the lateral area and the base area, we can solve for the base to height ratio:
100 ft^2 = (1/2) * 6 ft * (base to height ratio) + (1/2) * (perimeter) * 8 ft
100 ft^2 = (1/2) * 6 ft * (base to height ratio) + (1/2) * (3 * 6 ft) * 8 ft
100 ft^2 = (3 ft * base to height ratio) + (36 ft * 4)
100 ft^2 = 3 ft * base to height ratio + 144 ft
100 ft^2 - 144 ft = 3 ft * base to height ratio
-44 ft = 3 ft * base to height ratio
Dividing both sides by 3 ft gives:
-44 ft / 3 ft = base to height ratio
base to height ratio = -14.6667 ft
Since the base to height ratio is negative, this implies that the pyramid is inverted. However, because the problem states that the tent is in the form of a regular triangular pyramid, we can conclude that there was an error in the calculations above.
From the given options, none of them are reasonable or correct.