The surface area \( S \) of a square pyramid can be calculated using the formula:
\[ S = B + \frac{1}{2} P l \]
where:
- \( B \) is the area of the base,
- \( P \) is the perimeter of the base,
- \( l \) is the slant height of the pyramid.
Step 1: Calculate the area of the base \( B \): Since the base is a square with edge length \( a \):
\[ B = a^2 = 5^2 = 25 , \text{square yards} \]
Step 2: Calculate the perimeter \( P \): The perimeter \( P \) of a square is calculated as:
\[ P = 4a = 4 \times 5 = 20 , \text{yards} \]
Step 3: Calculate the slant height \( l \): The slant height can be found using the Pythagorean theorem. We can consider a right triangle where one leg is half the base length, the other leg is the height of the pyramid, and the hypotenuse is the slant height.
\[ l = \sqrt{\left(\frac{a}{2}\right)^2 + h^2} = \sqrt{\left(\frac{5}{2}\right)^2 + 25^2} \]
Calculating the values:
\[ \frac{5}{2} = 2.5 \]
\[ l = \sqrt{(2.5)^2 + 25^2} = \sqrt{6.25 + 625} = \sqrt{631.25} \]
Calculating \( \sqrt{631.25} \):
\[ l \approx 25.1 , \text{yards} \]
Step 4: Calculate the surface area \( S \):
Substituting \( B \), \( P \), and \( l \) into the surface area formula:
\[ S = 25 + \frac{1}{2} \times 20 \times 25.1 \]
Calculating:
\[ S = 25 + 10 \times 25.1 = 25 + 251 = 276 , \text{square yards} \]
Therefore, rounding to the nearest whole number, the surface area of the square pyramid is:
\[ \boxed{276} , \text{square yards} \]