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?

Question

a. 9.3 ft.
b. 6.5 ft.
c. 372 ft.
d. 57.3 ft.

Answer 1

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.