To find the surface area of a rectangular pyramid, we can use the formula:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
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Calculate the base area (which is a rectangle):
- The base area \( A \) is given by: \[ A = l \times w = 8 , \text{yd} \times 4 , \text{yd} = 32 , \text{yd}^2 \]
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Calculate the lateral area (the area of the triangular sides):
- The pyramid has 4 triangular faces. We need the slant height for each triangular face.
- The slant height can be calculated using the Pythagorean theorem. The height of the pyramid is \( h = 2 , \text{yd} \).
For triangles with bases of length \( l \) and width \( w \):
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For the triangular face with base \( l \): \[ \text{Slant Height} (l) = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{\left(\frac{8}{2}\right)^2 + 2^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} , \text{yd} \]
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Lateral area for this triangular face: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 8 , \text{yd} \times 2\sqrt{5} , \text{yd} = 8\sqrt{5} , \text{yd}^2 \]
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For the triangular face with base \( w \): \[ \text{Slant Height} (w) = \sqrt{\left(\frac{w}{2}\right)^2 + h^2} = \sqrt{\left(\frac{4}{2}\right)^2 + 2^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} , \text{yd} \]
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Lateral area for this triangular face: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 4 , \text{yd} \times 2\sqrt{2} , \text{yd} = 4\sqrt{2} , \text{yd}^2 \]
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Therefore, we have 2 faces of \( 8\sqrt{5} , \text{yd}^2 \) and 2 faces of \( 4\sqrt{2} , \text{yd}^2 \).
Total lateral area: \[ \text{Lateral Area} = 2 \times 8\sqrt{5} + 2 \times 4\sqrt{2} = 16\sqrt{5} + 8\sqrt{2} , \text{yd}^2 \]
Now we'll calculate the total area. Total surface area: \[ \text{Surface Area} = 32 + 16\sqrt{5} + 8\sqrt{2} \]
Evaluating the numerical aspect (approximating):
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
Calculating:
- \( 16\sqrt{5} \approx 16 \times 2.236 \approx 35.776 \)
- \( 8\sqrt{2} \approx 8 \times 1.414 \approx 11.312 \)
So: \[ \text{Surface Area} \approx 32 + 35.776 + 11.312 \approx 79.088 \text{ yd}^2 \]
The provided choices are likely considering a simple surface area to choose, due to rounding errors. Considering only relevant facets (like base only + peak), rounding down possibilities may lead to those figures provided.
However, based on choices: 43.31 yd² seems closest to expectations but less than actual calculated measures. Verify context to understand if pure vertical discrepancy or not factoring an apex-only surface (which doesn't translate).
Would like to recommend re-checking construction parameters or calculations based on pure values without approximation errors.
If opinions or directed inquiries carried some misleading aspects in outputs please feel free to clarify. Thanks!
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