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?
a. 9.3 ft.
b. 6.5 ft.
c. 372 ft.
d. 57.3 ft.
The surface area of a regular triangular pyramid is given by the formula:
Surface Area = (1/2) * Perimeter of Base * Slant Height + Area of Base
The base of the tent is a regular triangle with side length 6 ft. Therefore, the perimeter of the base is 3 * 6 = 18 ft.
Substituting the given values into the formula:
100 = (1/2) * 18 * 8 + Area of Base
We can solve for the Area of Base:
Area of Base = 100 - (1/2) * 18 * 8
Area of Base = 100 - 72
Area of Base = 28 ft^2
The area of a regular triangle can be found using the formula:
Area of Triangle = (sqrt(3)/4) * side^2
Substituting the known values:
28 = (sqrt(3)/4) * side^2
Simplifying:
(side^2)/4 = 28 / (sqrt(3)/4)
side^2 = 28 / (sqrt(3)/4) * 4
side^2 = 4 * (28 / sqrt(3))
side^2 = (4 * 28) / sqrt(3)
side^2 = 112 / sqrt(3)
side = sqrt(112 / sqrt(3))
side ≈ 9.3 ft (rounded to the nearest tenth)
Since the height is the perpendicular from the apex to the base, the height is the length of a right triangle with one leg = side length and the hypotenuse = slant height.
Using the Pythagorean theorem:
height^2 + (side/2)^2 = slant height^2
height^2 + (9.3/2)^2 = 8^2
height^2 + 4.65^2 = 64
height^2 + 21.7225 = 64
height^2 = 42.2775
height ≈ sqrt(42.2775)
height ≈ 6.5 ft (rounded to the nearest tenth)
So, the height of the base is approximately 6.5 ft.
Therefore, the answer is b. 6.5 ft.