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)

Question

Responses

9.3 ft.
9.3 ft.

372 ft.
372 ft.

6.5 ft.
6.5 ft.

57.3 ft.

Answer 1

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.

Answer 2

try again

Answer 3

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.

Answer 4

Choose a answer

Answer 5

Based on the calculations provided, the nearest tenth for the height of the base would be 7.4 ft. However, none of the given response options match this calculation.