To find the slant height of a square pyramid given the surface area and the length of the base, we can use the formula for the surface area of a square pyramid:
\[ SA = B + \frac{1}{2} \cdot P \cdot l \]
Where:
- \( SA \) is the surface area.
- \( B \) 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 a length of 40 inches, \[ B = \text{length}^2 = 40^2 = 1600 , \text{in}^2 \]
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Calculate the perimeter \( P \): For a square pyramid, the perimeter \( P \) of the base is given by: \[ P = 4 \cdot \text{length} = 4 \cdot 40 = 160 , \text{in} \]
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Substitute the values into the surface area formula: We know that \( SA = 2800 , \text{in}^2 \), so we can substitute the values into the equation: \[ 2800 = 1600 + \frac{1}{2} \cdot 160 \cdot l \]
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Simplify and solve for the slant height \( l \): First, subtract the area of the base from both sides: \[ 2800 - 1600 = \frac{1}{2} \cdot 160 \cdot l \] \[ 1200 = \frac{1}{2} \cdot 160 \cdot l \] Multiply both sides by 2: \[ 2400 = 160 \cdot l \] Now, divide both sides by 160: \[ l = \frac{2400}{160} = 15 , \text{in} \]
Thus, the slant height \( l \) is 15 inches.