To find the surface area of a rectangular pyramid, we need to calculate the area of the rectangular base and the area of the four triangular faces.
Given:
- Length (l) = 8 cm
- Width (w) = 4 cm
- Height (h) = 2 cm
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Area of the base (A_base): \[ A_{base} = l \times w = 8 , \text{cm} \times 4 , \text{cm} = 32 , \text{cm}^2 \]
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Area of the triangular faces: Each triangular face has a base equal to either the length or width of the pyramid, and the height of each triangle can be found using the slant height.
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Slant height (l_s) of the triangles: To find the slant height, we can use the Pythagorean theorem. For the triangular face with base equal to the length (l = 8 cm), the height of the triangle is h and half the base is \( \frac{l}{2} = \frac{8}{2} = 4 \) cm.
Using the Pythagorean theorem: \[ l_s = \sqrt{h^2 + \left(\frac{l}{2}\right)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.47 , \text{cm} \]
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The area of the two triangular faces with base length \( l = 8 , \text{cm} \): \[ A_{triangles_l} = 2 \times \left(\frac{1}{2} \times base \times height\right) = 2 \times \left(\frac{1}{2} \times 8 \times 4.47\right) = 2 \times (4 \times 4.47) = 2 \times 17.88 \approx 35.76 , \text{cm}^2 \]
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Calculate the slant height for the triangular face with base width (w = 4 cm): The half base is \( \frac{w}{2} = \frac{4}{2} = 2 \) cm: \[ l_s = \sqrt{h^2 + \left(\frac{w}{2}\right)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 , \text{cm} \]
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The area of the two triangular faces with base width \( w = 4 , \text{cm} \): \[ A_{triangles_w} = 2 \times \left(\frac{1}{2} \times 4 \times 2.83\right) = 2 \times (2 \times 2.83) = 2 \times 5.66 \approx 11.32 , \text{cm}^2 \]
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Total surface area (SA): \[ SA = A_{base} + A_{triangles_l} + A_{triangles_w} = 32 + 35.76 + 11.32 \approx 79.08 , \text{cm}^2 \]
Rounding to the nearest hundredth, the surface area is:
\[ \boxed{79.08} , \text{cm}^2 \]