To find the length of the edge of the square base of the pyramid using the Pythagorean theorem in three dimensions, you can imagine a right triangle where:
- The height of the pyramid (321 ft) is one leg.
- Half the length of the edge of the base (let's call this \( \frac{l}{2} \)) is the other leg.
- The slant height (436.5 ft) is the hypotenuse.
Applying the Pythagorean theorem:
\[ (\text{slant height})^2 = (\text{height})^2 + \left(\frac{l}{2}\right)^2 \]
Plugging in the numbers:
\[ (436.5)^2 = (321)^2 + \left(\frac{l}{2}\right)^2 \]
Calculating the squares:
\[ 190,479.25 = 103,041 + \left(\frac{l}{2}\right)^2 \]
Now, isolate \( \left(\frac{l}{2}\right)^2 \):
\[ \left(\frac{l}{2}\right)^2 = 190,479.25 - 103,041 \] \[ \left(\frac{l}{2}\right)^2 = 87,438.25 \]
Now, take the square root to find \( \frac{l}{2} \):
\[ \frac{l}{2} = \sqrt{87,438.25} \approx 295.6 \]
To find \( l \), multiply by 2:
\[ l = 2 \times 295.6 \approx 591.2 \]
Therefore, the length of the edge of the square base, rounded to the nearest tenth, is approximately:
591.2 feet