To find the length of the edge of the square base of the pyramid, we can use the relationship between the height of the pyramid, the slant height, and half the length of the edge of the base.
Let \( l \) be the length of the edge of the square base. The following relationships hold based on the right triangle formed by the height of the pyramid, the half length of the base, and the slant height:
- Height of pyramid = 321 ft
- Slant height = 436.5 ft
The half-length of the base is \( \frac{l}{2} \).
Using the Pythagorean theorem, we have:
\[ \text{(slant height)}^2 = (\text{height})^2 + \left(\frac{l}{2}\right)^2 \]
Plugging in the known values:
\[ (436.5)^2 = (321)^2 + \left(\frac{l}{2}\right)^2 \]
Calculating each term:
\[ (436.5)^2 = 190,546.25 \] \[ (321)^2 = 103,041 \]
Now substituting these values into the equation:
\[ 190,546.25 = 103,041 + \left(\frac{l}{2}\right)^2 \]
Next, we isolate \( \left(\frac{l}{2}\right)^2 \):
\[ \left(\frac{l}{2}\right)^2 = 190,546.25 - 103,041 \] \[ \left(\frac{l}{2}\right)^2 = 87,505.25 \]
Taking the square root gives:
\[ \frac{l}{2} = \sqrt{87,505.25} \] \[ \frac{l}{2} \approx 295.8 \]
Now, we multiply by 2 to find \( l \):
\[ l \approx 595.6 \]
Thus, the length of the edge of the square base is approximately 595.6 feet.