To calculate the surface area of a rectangular pyramid, we need to consider both the area of the base and the area of the triangular sides (lateral surface area).
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Calculate the area of the base: The base is a rectangle with lengths of 10 ft and 7 ft. \[ \text{Area of base} = \text{length} \times \text{width} = 10 , \text{ft} \times 7 , \text{ft} = 70 , \text{ft}^2 \]
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Calculate the area of the triangular sides: The pyramid has four triangular faces, two corresponding to each side of the base.
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Triangles corresponding to the longer side (10 ft):
- The slant height for this side (which is the height of the triangle) is 7 ft.
- Each triangular face has a base of 10 ft.
- Area of one triangle = \( \frac{1}{2} \times \text{base} \times \text{height} \) \[ \text{Area of one triangle} = \frac{1}{2} \times 10 , \text{ft} \times 7 , \text{ft} = 35 , \text{ft}^2 \]
- Total area for the two triangles on the longer side: \[ 2 \times 35 , \text{ft}^2 = 70 , \text{ft}^2 \]
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Triangles corresponding to the shorter side (7 ft):
- The slant height for this side is 8 ft.
- Each triangular face has a base of 7 ft. \[ \text{Area of one triangle} = \frac{1}{2} \times 7 , \text{ft} \times 8 , \text{ft} = 28 , \text{ft}^2 \]
- Total area for the two triangles on the shorter side: \[ 2 \times 28 , \text{ft}^2 = 56 , \text{ft}^2 \]
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Calculate total lateral surface area: \[ \text{Total lateral area} = \text{Area from longer side} + \text{Area from shorter side} = 70 , \text{ft}^2 + 56 , \text{ft}^2 = 126 , \text{ft}^2 \]
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Calculate the total surface area of the pyramid: \[ \text{Surface area} = \text{Area of base} + \text{Total lateral area} = 70 , \text{ft}^2 + 126 , \text{ft}^2 = 196 , \text{ft}^2 \]
Thus, the total surface area of the pyramid is \(\boxed{196 , \text{ft}^2}\).
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