To find the surface area of a square pyramid, we can use the formula:
\[ \text{Surface Area} = B + \frac{1}{2} P l \]
where:
- \( B \) is the area of the base,
- \( P \) is the perimeter of the base,
- \( l \) is the slant height of the pyramid.
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Calculate the area of the base \( B \): Since the base is a square with edge length \( a = 5 \) yards, the area \( B \) is given by: \[ B = a^2 = 5^2 = 25 \text{ square yards} \]
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Calculate the perimeter \( P \): The perimeter of the base, which is a square, is given by: \[ P = 4a = 4 \times 5 = 20 \text{ yards} \]
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Calculate the slant height \( l \): The slant height can be found using the Pythagorean theorem. The height \( h \) is 25 yards, and the half of the base edge length is \( \frac{5}{2} = 2.5 \) yards. Thus: \[ l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2} = \sqrt{25^2 + 2.5^2} = \sqrt{625 + 6.25} = \sqrt{631.25} \approx 25.1 \text{ yards} \]
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Calculate the lateral surface area: The lateral surface area \( \frac{1}{2} P l \) is: \[ \frac{1}{2} \times 20 \times 25.1 \approx 252 \text{ square yards} \]
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Calculate the total surface area: Now, add the area of the base and the lateral surface area: \[ \text{Surface Area} = B + \frac{1}{2} P l \approx 25 + 252 = 277 \text{ square yards} \]
Rounding to the nearest whole number, the surface area is approximately 277 square yards.
Thus, the best answer from the options given would be: 276 square yards.