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?(1 point)
Responses
9.3 ft.
9.3 ft.
372 ft.
372 ft.
6.5 ft.
6.5 ft.
57.3 ft.
The surface area of a triangular pyramid can be calculated using the formula:
Surface Area = base area + lateral area
The base area of a triangular pyramid is given by the formula:
Base Area = (base length * base length * √3) / 4
In this case, the base length is given as 6 ft, so we can substitute this into the formula:
Base Area = (6 * 6 * √3) / 4
Base Area = 9√3 ft.²
The lateral area of a triangular pyramid is given by the formula:
Lateral Area = (base length * slant height) / 2
Substituting the given values:
Lateral Area = (6 * 8) / 2
Lateral Area = 24 ft.²
Now, we can calculate the height of the pyramid using the formula for surface area:
Surface Area = Base Area + Lateral Area
Substituting the known values:
100 = 9√3 + 24
Subtracting 24 from both sides:
76 = 9√3
Dividing both sides by 9:
8.44 = √3
Squaring both sides to remove the square root:
(8.44)² = (√3)²
71.2336 ≈ 3
Now, we can find the height of the pyramid using the Pythagorean theorem:
h^2 = slant height^2 - base length^2
h^2 = 8^2 - 6^2
h^2 = 64 - 36
h^2 = 28
h ≈ √28
h ≈ 5.3 ft
Therefore, the height of the base to the nearest tenth is 5.3 ft. None of the response options provided match this calculation.
try again
I apologize for the error. Let's try again:
The surface area of a triangular pyramid is given by:
Surface Area = base area + lateral area
The base area of a triangular pyramid is given by the formula:
Base Area = (base length * base length * √3) / 4
In this case, the base length is given as 6 ft, so we can substitute this into the formula:
Base Area = (6 * 6 * √3) / 4
Base Area = 9√3 ft.²
The lateral area of a triangular pyramid is given by the formula:
Lateral Area = (base length * slant height) / 2
Substituting the given values:
Lateral Area = (6 * 8) / 2
Lateral Area = 24 ft.²
Now, we can calculate the height of the pyramid using the formula for surface area:
Surface Area = Base Area + Lateral Area
Substituting the known values:
100 = 9√3 + 24
Subtracting 24 from both sides:
76 = 9√3
Dividing both sides by 9:
8.44 = √3
Squaring both sides to remove the square root:
(8.44)² = (√3)²
71.2336 ≈ 3
Now, we can find the height of the pyramid. Let h represent the height of the pyramid.
Using the Pythagorean theorem:
h^2 = slant height^2 - base length^2
h^2 = 8^2 - 3^2
h^2 = 64 - 9
h^2 = 55
h ≈ √55
h ≈ 7.416198487 ft
Therefore, the height of the base to the nearest tenth is approximately 7.4 ft. None of the response options provided match this calculation.