To calculate the surface area of the rectangular pyramid, we need to find the area of the base and the areas of the four triangular faces.
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Base Area: The area of the base is given by the formula: \[ \text{Base Area} = \text{length} \times \text{width} = 10 , \text{ft} \times 7 , \text{ft} = 70 , \text{ft}^2 \]
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Area of the Triangles: The pyramid has two triangular faces with slant heights along the longer side of the base (10 ft) and two with slant heights along the shorter side of the base (7 ft).
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Triangles parallel to the longer side (10 ft):
- Each triangle has a base of 7 ft and a slant height of 8 ft.
- Area of one triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 , \text{ft} \times 8 , \text{ft} = 28 , \text{ft}^2 \]
- Since there are two such triangles: \[ \text{Total Area for longer side} = 2 \times 28 , \text{ft}^2 = 56 , \text{ft}^2 \]
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Triangles parallel to the shorter side (7 ft):
- Each triangle has a base of 10 ft and a slant height of 7 ft.
- Area of one triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 , \text{ft} \times 7 , \text{ft} = 35 , \text{ft}^2 \]
- Since there are two such triangles: \[ \text{Total Area for shorter side} = 2 \times 35 , \text{ft}^2 = 70 , \text{ft}^2 \]
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Total Surface Area: Now we can add the area of the base and the areas of the four triangles: \[ \text{Total Surface Area} = \text{Base Area} + \text{Area from longer sides} + \text{Area from shorter sides} \] \[ \text{Total Surface Area} = 70 , \text{ft}^2 + 56 , \text{ft}^2 + 70 , \text{ft}^2 = 196 , \text{ft}^2 \]
Thus, the correct calculation for the surface area of the pyramid is 196 ft² or 196 ft. squared.