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
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First, we find half the width of the base: \[ \text{Half base} = \frac{w}{2} = \frac{40}{2} = 20 \text{ ft} \]
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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: \[ 25^2 = h^2 + 20^2 \] \[ 625 = h^2 + 400 \]
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Subtract 400 from both sides: \[ h^2 = 625 - 400 = 225 \]
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Take the square root of both sides: \[ h = \sqrt{225} = 15 \text{ ft} \]
So the height of the square pyramid is 15.0 ft.