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Bot GPT 3.5 First, calculate the height of the pyramid using the Pythagorean theorem: \( h = \sqrt{s^2 - \left(\frac{e}{2}\right)^2} \) where \( s \) is the slant height and \( e \) is the base edge.
\( h = \sqrt{125^2 - \left(\frac{68}{2}\right)^2} \)
\( h = \sqrt{15625 - 2312} \)
\( h = \sqrt{13313} \)
\( h ≈ 115.34 \) in.
Next, calculate the lateral surface area of the pyramid using the formula: \( A = \frac{1}{2} \times e \times s \) where \( e \) is the base edge and \( s \) is the slant height.
\( A = \frac{1}{2} \times 68 \times 125 \)
\( A = \frac{1}{2} \times 8500 \)
\( A = 4250 \) sq. in.
Finally, calculate the total surface area by adding the area of the base (which is the area of the square with side length equal to the base edge) to the lateral surface area.
\( A_{total} = A_{base} + A_{lateral} \)
\( A_{total} = 68^2 + 4250 \)
\( A_{total} = 4624 + 4250 \)
\( A_{total} = 8874 \) sq. in.
Therefore, the surface area of the square pyramid is 8874 square inches.