To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the area of the four triangular faces.
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Calculate the area of the base:
The base of the pyramid is a rectangle with length \( l \) and width \( w \).\[ \text{Area of the base} = l \times w = 15 , \text{m} \times 12 , \text{m} = 180 , \text{m}^2 \]
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Calculate the area of the triangular faces:
Each triangular face has a base on the side of the rectangle and a height that can be found using the given height of the pyramid \( h \) and the length of half the base.The triangular faces can be categorized as follows:
- Two triangular faces with a base equal to \( l \) (length)
- Two triangular faces with a base equal to \( w \) (width)
The slant height of the pyramid can be calculated for each face using the Pythagorean theorem. We need to find the distance from the apex of the pyramid directly to the midpoint of the base side (which will be the height of the triangle).
For the two triangular faces with base \( l \):
- The midpoint of the base \( l \) is \( \frac{l}{2} = \frac{15}{2} = 7.5 , \text{m} \).
- Thus, the height \( h_t \) of the triangular face can be calculated as:
\[ h_t = \sqrt{h^2 + \left(\frac{l}{2}\right)^2} = \sqrt{10^2 + 7.5^2} = \sqrt{100 + 56.25} = \sqrt{156.25} = 12.5 , \text{m} \]
The area of one triangular face with base \( l \):
\[ A_l = \frac{1}{2} \times l \times h_t = \frac{1}{2} \times 15 \times 12.5 = 93.75 , \text{m}^2 \]
Since there are two triangular faces of this type:
\[ \text{Total area for \( l \)} = 2 \times 93.75 = 187.5 , \text{m}^2 \]
For the two triangular faces with base \( w \):
- The midpoint of the base \( w \) is \( \frac{w}{2} = \frac{12}{2} = 6 , \text{m} \).
The height \( h_t \) for these faces is:
\[ h_t = \sqrt{h^2 + \left(\frac{w}{2}\right)^2} = \sqrt{10^2 + 6^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.62 , \text{m} \]
The area of one triangular face with base \( w \):
\[ A_w = \frac{1}{2} \times w \times h_t = \frac{1}{2} \times 12 \times 11.62 \approx 69.72 , \text{m}^2 \]
Since there are two triangular faces of this type:
\[ \text{Total area for \( w \)} = 2 \times 69.72 \approx 139.44 , \text{m}^2 \]
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Calculate the total surface area:
Now we add the area of the base and the areas of the triangular faces together:
\[ \text{Total Surface Area} = \text{Area of the base} + \text{Total area for \( l \)} + \text{Total area for \( w \)} \]
\[ \text{Total Surface Area} = 180 + 187.5 + 139.44 \approx 506.94 , \text{m}^2 \]
Thus, the surface area of the rectangular pyramid is approximately:
\[ \boxed{506.94} , \text{m}^2 \]