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.", the base is 6 ft., and the slant height is 8 ft. What is the height of the base to the nearest tenth?
The surface area of a triangular pyramid is given by the formula S = 1/2 * (base perimeter) * slant height.
Since the tent is a regular triangular pyramid, all three sides of the base are equal. Therefore, the base perimeter is 3 times the length of one side of the base.
Let's call the length of one side of the base x. So, the base perimeter is 3x.
We know that the surface area of the tent is 100 ft² and the base is 6 ft. So we have:
100 = 0.5 * 3x * 8.
Simplifying this equation, we have:
100 = 12x.
Dividing both sides of the equation by 12, we find:
x = 100/12 = 25/3 ≈ 8.33 ft.
The height of the tent is given by the formula: height = sqrt(slant height^2 - (base side length/2)^2).
Substituting the given values, we have:
height = sqrt(8^2 - (8.33/2)^2)
= sqrt(64 - (4.17)^2)
= sqrt(64 - 17.34)
= sqrt(46.66)
≈ 6.83 ft.
Therefore, the height of the base of the tent is approximately 6.8 ft.