To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the area of the triangular faces.
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Area of the Base (rectangle): \[ \text{Area}_{\text{base}} = l \times w = 15 , \text{m} \times 12 , \text{m} = 180 , \text{m}^2 \]
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Area of the Triangular Faces: There are four triangular faces. We need to find the area of each triangular face. The height of each triangular face is calculated using the slant height of the pyramid, which we find using the Pythagorean theorem.
First, calculate the slant height for each pair of triangles and their base:
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For the triangles on the length sides (15 m):
- The half of the base length = \(\frac{l}{2} = \frac{15}{2} = 7.5 , \text{m}\)
- The height of the pyramid \(h = 10 , \text{m}\)
- The slant height \(s\) can be found as follows: \[ s = \sqrt{(h)^2 + \left(\frac{l}{2}\right)^2} = \sqrt{10^2 + 7.5^2} = \sqrt{100 + 56.25} = \sqrt{156.25} = 12.5 , \text{m} \]
- Area of one triangular face: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 15 \times 12.5 = 93.75 , \text{m}^2 \]
- There are two such triangular faces, so combined area: \[ \text{Total area for length sides} = 2 \times 93.75 = 187.5 , \text{m}^2 \]
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For the triangles on the width sides (12 m):
- The half of the base width = \(\frac{w}{2} = \frac{12}{2} = 6 , \text{m}\)
- The height of the pyramid is still \(h = 10 , \text{m}\)
- The slant height \(s\) is calculated as follows: \[ s = \sqrt{(h)^2 + \left(\frac{w}{2}\right)^2} = \sqrt{10^2 + 6^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.62 , \text{m} \]
- Area of one triangular face: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 11.62 \approx 69.72 , \text{m}^2 \]
- There are two such triangular faces, so combined area: \[ \text{Total area for width sides} = 2 \times 69.72 \approx 139.44 , \text{m}^2 \]
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Total Surface Area: Now combine all areas: \[ \text{Total Surface Area} = \text{Area}_{\text{base}} + \text{Total area for length sides} + \text{Total area for width sides} \] \[ = 180 + 187.5 + 139.44 \approx 506.94 , \text{m}^2 \]
Rounding to the nearest hundredth: \[ \text{Total Surface Area} \approx 506.94 , \text{m}^2 \]
Thus, the final answer is: \[ \boxed{506.94} , \text{m}^2 \]