To calculate the surface area of the triangular prism shaped tent, we need to consider both the lateral area of the prism and the area of the triangular bases.
Given dimensions:
- Height (height of the triangular face): \( h = 9 , \text{ft} \)
- Base (width of the triangular face): \( b = 20 , \text{ft} \)
- Length of the prism: \( l = 24 , \text{ft} \)
- Hypotenuse of the triangular face: \( c = 13.45 , \text{ft} \)
Step 1: Calculate the area of the triangular base
The area \( A \) of a triangle is calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting in the values we have:
\[ A = \frac{1}{2} \times 20 \times 9 = 90 , \text{ft}^2 \]
Step 2: Calculate the lateral surface area of the prism
The lateral surface area \( A_L \) of the prism consists of three rectangular sides:
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Two triangles on the left and right (each side area is the height of the prism times the length): \[ A_{\text{triangular bases}} = 2 \times A = 2 \times 90 = 180 , \text{ft}^2 \]
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Two rectangles for the two remaining faces:
- The area of the rectangular faces (width times length): \[ A_{\text{rectangles}} = 2 \times (l \times b) = 2 \times (24 \times 20) = 960 , \text{ft}^2 \]
Step 3: Calculate the total surface area
Adding both the areas of the triangular bases and the rectangular faces together will give us the total surface area \( SA \):
\[ SA = A_{\text{triangular bases}} + A_{\text{rectangles}} = 180 + 960 = 1140 , \text{ft}^2 \]
Conclusion
The calculated surface area does not match any of the provided options. However, if we consider that one dimension or area calculation may misalign due to different interpretations of the shape, we can also consider that:
If we analyze this based on a simplified triangular prism with only external faces, it might yield a closer approach to one of the options provided. Despite that, an appropriate design with these calculations seems essential for clarification.
Given the closest value from the provided response options and to match with a common outlay of triangular prism areas, rounding might suggest an answer near the closest response.
From the choices, it’s uncertain but if asked to choose, consider:
- 982.8 ft² or 1,215.60 ft² as the first nearest, but the right approach may not yield a precise decision from the calculations above.