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.
57.3 ft.
57.3 ft.
372 ft.
372 ft.
9.3 ft.
The formula for the surface area of a regular triangular pyramid is given by:
Surface Area = base area + lateral area
The base area of a regular triangular pyramid is given by:
Base area = (1/2) * base * height
And the lateral area is given by:
Lateral area = (1/2) * perimeter * slant height
Given that the surface area is 100 ft^2, the base is 6 ft, and the slant height is 8 ft, we can set up the equation:
100 = (1/2) * 6 * height + (1/2) * perimeter * 8
Since it is a regular triangular pyramid, the perimeter is equal to 3 times the base. Therefore, we have:
100 = (1/2) * 6 * height + (1/2) * 3 * 6 * 8
100 = 3 * height + 3 * 6 * 8
100 = 3 * height + 144
Subtracting 144 from both sides gives:
-44 = 3 * height
Dividing both sides by 3 gives:
-44/3 = height
Therefore, the height of the base to the nearest tenth is -14.7 ft. However, since the height of a pyramid cannot be negative, we can conclude that the given values for the surface area, base, and slant height do not form a valid triangular pyramid. Therefore, none of the provided options (6.5 ft, 57.3 ft, 372 ft, or 9.3 ft) is correct.