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? (1 point)
The surface area of a regular triangular pyramid is given by the formula:
Surface Area = Base Area + (1/2) * Perimeter * Slant Height
First, let's find the base area. The base is in the shape of a regular triangle, so the formula for the area of an equilateral triangle is:
Base Area = (sqrt(3) / 4) * side^2
In this case, the side of the triangle is 6 ft., so:
Base Area = (sqrt(3) / 4) * 6^2 = (sqrt(3) / 4) * 36 = 9 * sqrt(3) ft^2
Now let's find the perimeter of the base. Since the base is a regular triangle, all the sides are equal, so the perimeter is 3 times the length of one side:
Perimeter = 3 * side = 3 * 6 = 18 ft.
Now we can substitute these values into the surface area formula and solve for the height:
100 ft^2 = 9 * sqrt(3) ft^2 + (1/2) * 18 ft. * 8 ft. * height
Simplifying:
100 ft^2 = 9 * sqrt(3) ft^2 + 72 ft. * height
Subtracting 9 * sqrt(3) ft^2 from both sides:
100 ft^2 - 9 * sqrt(3) ft^2 = 72 ft. * height
Simplifying:
100 ft^2 - 9 * sqrt(3) ft^2 = 72 ft. * height
72 ft. * height = 100 ft^2 - 9 * sqrt(3) ft^2
Dividing both sides by 72 ft.:
height = (100 ft^2 - 9 * sqrt(3) ft^2) / 72 ft.
height ≈ (100 - 9 * sqrt(3)) / 72
height ≈ (100 - 9 * 1.732) / 72
height ≈ (100 - 15.588) / 72
height ≈ 84.412 / 72
height ≈ 1.173 ft.
Therefore, the height of the base to the nearest tenth is 1.2 ft.