To determine the height of the square pyramid using the Pythagorean theorem, we can use the relationship involving the height, the slant height, and half the length of the base.
Given:
- \( w \) (the width of the square base) = 40 ft
- \( s \) (the slant height) = 25 ft
1. First, we find half the width of the base:
\[
\text{Half base} = \frac{w}{2} = \frac{40}{2} = 20 \text{ ft}
\]
2. Now, we apply the Pythagorean theorem. In a right triangle formed by the height of the pyramid (h), half the base (20 ft), and the slant height (25 ft), we have:
\[
s^2 = h^2 + \left(\frac{w}{2}\right)^2
\]
Plugging in the values we have:
\[
25^2 = h^2 + 20^2
\]
\[
625 = h^2 + 400
\]
3. Subtract 400 from both sides:
\[
h^2 = 625 - 400 = 225
\]
4. Take the square root of both sides:
\[
h = \sqrt{225} = 15 \text{ ft}
\]
So the height of the square pyramid is **15.0 ft**.