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)
• 6.5 ft.
• 57.3 ft.
• 9.3 ft.
• 372 ft.

1 answer

To find the height of the base of the triangular pyramid (tent), we first need to understand the surface area and the geometric properties of a triangular pyramid.

The surface area \( S \) of a triangular pyramid can be calculated using the formula:

\[ S = \text{Base Area} + \text{Lateral Area} \]

Step 1: Calculate the area of the base

The base of the pyramid is a triangle, and we know the base length and need to find the height corresponding to this base. The area \( A \) of the triangular base can be calculated using:

\[ A = \frac{1}{2} \times \text{base} \times \text{height of the triangle} \]

Given that the base \( b = 6 \) ft, we denote the height of the triangle as \( h_b \).

Step 2: Find the lateral area

The slant height \( l \) provided is 8 ft. The lateral area of the pyramid consists of three triangular faces, each having a base of 6 ft and a height equal to the slant height.

The area of one triangular face is given by:

\[ A_{\text{face}} = \frac{1}{2} \times \text{base} \times \text{slant height} \]

Given each triangular face has a base of 6 ft and a height (slant height) of 8 ft, the area of one face is:

\[ A_{\text{face}} = \frac{1}{2} \times 6 \times 8 = 24 , \text{ft}^2 \]

Since there are three such faces, the total lateral area \( A_{\text{lateral}} \) is:

\[ A_{\text{lateral}} = 3 \times A_{\text{face}} = 3 \times 24 = 72 , \text{ft}^2 \]

Step 3: Calculate the area of the base

So, we can equate the total surface area to the sum of the base area and the lateral area:

\[ 100 = A_{\text{base}} + 72 \]

Solving for the base area:

\[ A_{\text{base}} = 100 - 72 = 28 , \text{ft}^2 \]

Now, using the formula for the area of the base:

\[ 28 = \frac{1}{2} \times 6 \times h_b \]

Solving for \( h_b \):

\[ 28 = 3 \times h_b \] \[ h_b = \frac{28}{3} \approx 9.3 , \text{ft} \]

Thus, the height of the base to the nearest tenth is approximately:

9.3 ft.