The formula for the surface area \( SA \) of a square pyramid is given by:
\[ SA = \frac{1}{2} \times P \times l + B \]
where:
- \( P \) is the perimeter of the base,
- \( l \) is the slant height,
- \( B \) is the area of the base.
For a square base with side length \( s \):
\[ P = 4s \] \[ B = s^2 \]
In this case, the side length \( s \) is given as 40 inches.
Step 1: Calculate the perimeter and base area. \[ P = 4s = 4 \times 40 = 160 \text{ in} \] \[ B = s^2 = 40^2 = 1600 \text{ in}^2 \]
Step 2: Substitute into the surface area formula. \[ SA = \frac{1}{2} \times P \times l + B \] \[ 2800 = \frac{1}{2} \times 160 \times l + 1600 \]
Step 3: Simplify and solve for \( l \). First, subtract the base area \( B \) from both sides: \[ 2800 - 1600 = \frac{1}{2} \times 160 \times l \] \[ 1200 = 80l \]
Now divide both sides by 80: \[ l = \frac{1200}{80} = 15 \text{ in} \]
Thus, the slant height \( l \) is \( 15 \) inches.
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