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.
For a square base:
- The area of the base, \( B \), is \( s^2 \) where \( s \) is the length of the base.
- The perimeter of the base, \( P \), is \( 4s \).
Given:
- The length of the base \( s = 40 \) in.
- Surface area \( SA = 2800 \) in².
First, calculate the base area \( B \): \[ B = s^2 = 40^2 = 1600 \text{ in}^2 \]
Now, calculate the perimeter \( P \): \[ P = 4s = 4 \cdot 40 = 160 \text{ in} \]
Now, substitute the values into the surface area formula: \[ 2800 = 1600 + \frac{1}{2} \cdot 160 \cdot l \]
Subtract \( 1600 \) from both sides: \[ 2800 - 1600 = \frac{1}{2} \cdot 160 \cdot l \] \[ 1200 = 80l \]
Now, solve for \( l \): \[ l = \frac{1200}{80} = 15 \text{ in} \]
Therefore, the slant height \( l \) is 15 in.
2 answers