To find the surface area \( S \) of a square pyramid, you can use the following formula:
\[ S = B + L \]
where:
- \( B \) is the area of the base,
- \( L \) is the lateral surface area.
For a square pyramid:
- The base edge \( a = 68 \) in.
The area of the base \( B \) is given by:
\[ B = a^2 = 68^2 = 4624 \text{ in}^2 \]
The lateral surface area \( L \) for a square pyramid is given by:
\[ L = \frac{1}{2} \times P \times l \]
where:
- \( P \) is the perimeter of the base,
- \( l \) is the slant height.
The perimeter \( P \) of the base (a square) is:
\[ P = 4a = 4 \times 68 = 272 \text{ in} \]
Now, substituting the values into the lateral surface area formula:
- The slant height \( l = 125 \) in.
\[ L = \frac{1}{2} \times 272 \times 125 = 34,000 \text{ in}^2 \]
Now, the total surface area \( S \) is:
\[ S = B + L = 4624 + 34,000 = 38,624 \text{ in}^2 \]
However, based on the structure of the responses provided, it seems we're likely looking for a mistake in the calculation. Let's verify the lateral area again more carefully.
Lateral area:
\[ L = \frac{1}{2} \times 272 \times 125 = 34,000 \text{ in}^2 \]
But recalculating the perimeter part shows possible error from description.
The lateral surface comes from: \[ L = \frac{1}{2}(4a)(l) \] Calculating: \[ L = 2 \cdot 68 \cdot 125 = 17000 \text{ in}^2 \] Now to use again: \[ S = B + L \] Where \( B = 4624 + 17000 = 21,624 \text{ in}^2\).
So, the correct answer is:
21,624 in²