To calculate the lateral surface area of a rectangular pyramid, we need to find the area of the four triangular faces that make up the sides of the pyramid.
The formula for the lateral surface area \( A_L \) of a pyramid is given by the sum of the areas of the triangular sides. For a rectangular pyramid, we can separate the faces based on the dimensions of the base.
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Identify the Dimensions:
- Base length (\( b_1 \)): 14 ft
- Base width (\( b_2 \)): 13 ft
- Slant height corresponding to base length (\( l_1 \)): 12 ft
- Slant height corresponding to base width (\( l_2 \)): 13 ft
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Calculate Areas of the Triangular Faces:
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Two triangular faces correspond to the longer side (base length of 14 ft):
- Area of one triangular face: \[ A_{triang1} = \frac{1}{2} \times b_1 \times l_1 = \frac{1}{2} \times 14 \times 12 = 84 \text{ ft}^2 \]
- So, total area for the two triangular faces corresponding to the longer side: \[ A_{triang1, total} = 2 \times 84 = 168 \text{ ft}^2 \]
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Two triangular faces correspond to the shorter side (base width of 13 ft):
- Area of one triangular face: \[ A_{triang2} = \frac{1}{2} \times b_2 \times l_2 = \frac{1}{2} \times 13 \times 13 = \frac{169}{2} = 84.5 \text{ ft}^2 \]
- So, total area for the two triangular faces corresponding to the shorter side: \[ A_{triang2, total} = 2 \times 84.5 = 169 \text{ ft}^2 \]
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Calculate the Total Lateral Surface Area: Combine the areas of all four triangular faces: \[ A_L = A_{triang1, total} + A_{triang2, total} = 168 + 169 = 337 \text{ ft}^2 \]
Thus, the lateral surface area of the pyramid is \( \mathbf{337 \text{ ft}^2} \).