To find the surface area of a square pyramid, you can use the formula:
\[ \text{Surface Area} = B + \frac{1}{2} \cdot P \cdot l \]
where:
- \( B \) is the area of the base,
- \( P \) is the perimeter of the base,
- \( l \) is the slant height.
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Calculate the area of the base \( B \): Since the base is a square with edge length of 5 yards: \[ B = \text{edge length}^2 = 5^2 = 25 \text{ square yards} \]
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Calculate the perimeter \( P \): The perimeter of a square is given by: \[ P = 4 \cdot \text{edge length} = 4 \cdot 5 = 20 \text{ yards} \]
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Calculate the slant height \( l \): The height of the pyramid is given as 25 yards. To find the slant height, you can use the Pythagorean theorem. The slant height \( l \) is the hypotenuse of a right triangle where one leg is half of the base edge length (which is \( \frac{5}{2} = 2.5 \) yards), and the other leg is the height of the pyramid (25 yards).
Thus: \[ l = \sqrt{(\text{height})^2 + \left(\frac{\text{base edge length}}{2}\right)^2} \] \[ l = \sqrt{25^2 + 2.5^2} = \sqrt{625 + 6.25} = \sqrt{631.25} \approx 25.14 \text{ yards} \]
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Now plug these values into the surface area formula: \[ \text{Surface Area} = B + \frac{1}{2} \cdot P \cdot l \] \[ \text{Surface Area} = 25 + \frac{1}{2} \cdot 20 \cdot 25.14 \] \[ \text{Surface Area} = 25 + 10 \cdot 25.14 = 25 + 251.4 = 276.4 \]
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Round to the nearest whole number: The surface area is approximately 276.4, which rounds to 276 square yards.
So, the final answer is 276 square yards.
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