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
The surface area of a regular triangular pyramid is given by the formula:
Surface Area = (1/2) x Perimeter of Base x Slant Height
Since the base of the pyramid is a regular triangle, the perimeter of the base is 3 times the length of any side. Let's call this length "s".
The formula for the surface area can be rewritten as:
100 = (1/2) x (3s) x 8
Multiplying both sides by 2 to get rid of the fraction:
200 = 24s
Dividing both sides by 24:
s = 200/24 = 25/3 ≈ 8.333...
Since we need to find the height of the base, we need to calculate the height perpendicular to the base of the pyramid.
Using the Pythagorean theorem, we can find the height "h" by finding the square root of (slant height^2 - (base/2)^2):
h = √(8^2 - (6/2)^2)
h = √(64 - 9)
h ≈ √55
h ≈ 7.416...
Therefore, the height of the base is approximately 7.4 ft to the nearest tenth.