To find the surface area of a rectangular pyramid, you need to calculate the area of its base and the areas of its four triangular faces.
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Calculate the area of the base: The base of the pyramid is a rectangle: \[ \text{Area}_{\text{base}} = l \times w = 15, \text{m} \times 12, \text{m} = 180, \text{m}^2 \]
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Calculate the area of the triangular faces: There are four triangular faces, two of them having base \(l\) and height \(h\) and two having base \(w\) and height \(h\).
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For the triangular faces with base \(l\) (15 m): The area for one triangular face is: \[ \text{Area}_{\text{triangular face}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 15 \times \sqrt{\left(\frac{w}{2}\right)^2 + h^2} \] where \(w = 12, \text{m}\), and \(h = 10, \text{m}\).
First, calculate the height of the triangular face: \[ \text{Height}_{\text{triangular face}} = \sqrt{\left(\frac{12}{2}\right)^2 + 10^2} = \sqrt{6^2 + 10^2} = \sqrt{36 + 100} = \sqrt{136} \approx 11.66, m \]
Now, calculate the area of one triangular face with base \(l\): \[ \text{Area}_{\text{triangular face (l)}} = \frac{1}{2} \times 15 \times 11.66 \approx 87.45, m^2 \] Since there are two such faces, the total area is: \[ 2 \times 87.45, m^2 \approx 174.90, m^2 \]
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For the triangular faces with base \(w\) (12 m): The area is calculated in the same way: \[ \text{Area}{\text{triangular face (w)}} = \frac{1}{2} \times 12 \times \sqrt{\left(\frac{15}{2}\right)^2 + 10^2} \] where the height is: \[ \text{Height}{\text{triangular face}} = \sqrt{\left(\frac{15}{2}\right)^2 + 10^2} = \sqrt{7.5^2 + 10^2} = \sqrt{56.25 + 100} = \sqrt{156.25} \approx 12.5, m \]
Now, calculate the area of one triangular face with base \(w\): \[ \text{Area}_{\text{triangular face (w)}} = \frac{1}{2} \times 12 \times 12.5 = 75, m^2 \] Since there are two such faces, the total area is: \[ 2 \times 75, m^2 = 150, m^2 \]
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Calculate the total surface area: \[ \text{Total Surface Area} = \text{Area}{\text{base}} + \text{Area}{\text{triangular faces}}(l) + \text{Area}_{\text{triangular faces}}(w) \] \[ \text{Total Surface Area} = 180, m^2 + 174.90, m^2 + 150, m^2 = 504.90, m^2 \]
Thus, the surface area of the rectangular pyramid is approximately: \[ \boxed{504.90}, m^2 \]