To find the height of the triangular pyramid, we will first calculate the height of the triangular base using the given information.
The surface area \( S \) of a triangular pyramid is given by the formula:
\[ S = \text{Base Area} + \text{Lateral Area} \]
- Calculate the area of the base:
The base of the triangular pyramid is a triangle with a base \( b = 6 \) ft. To find its height \( h_b \), we need to use the formula for the area of a triangle.
The area of the triangle can be expressed as: \[ \text{Area} = \frac{1}{2} \times b \times h_b \]
- Calculate the lateral surface area:
The lateral area of a triangular pyramid can be calculated as the sum of the areas of the three triangular faces. Each face has a base of \( b = 6 \) ft and a slant height \( l = 8 \) ft. The area of one triangular face is:
\[ \text{Area (face)} = \frac{1}{2} \times b \times l = \frac{1}{2} \times 6 \times 8 = 24 \text{ ft}^2 \]
Since there are three faces, the total lateral area \( LA \) is:
\[ LA = 3 \times 24 = 72 \text{ ft}^2 \]
- Find the area of the base:
The total surface area of the tent is \( 100 \text{ ft}^2 \). Therefore, we can write:
\[ 100 = A_{\text{base}} + LA \]
Substituting for \( LA \):
\[ 100 = A_{\text{base}} + 72 \]
Solving for the area of the base:
\[ A_{\text{base}} = 100 - 72 = 28 \text{ ft}^2 \]
- Solve for the height of the base \( h_b \):
Now we can use the area of the base to find the height \( h_b \): \[ A_{\text{base}} = \frac{1}{2} \times b \times h_b \implies 28 = \frac{1}{2} \times 6 \times h_b \]
Solving for \( h_b \):
\[ 28 = 3 h_b \implies h_b = \frac{28}{3} \approx 9.3 \text{ ft} \]
Hence, the height of the base to the nearest tenth is \( 9.3 \text{ ft} \).
Therefore, the answer is 9.3 ft.