To find the slant height of the square pyramid, we can use the Pythagorean theorem.
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Identify the values:
- The base of the pyramid is a square with each side measuring 40 miles.
- The height of the pyramid (vertical height from the base to the apex) is 9 miles.
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Find the length of half the base:
- Since the base is square, half of one side of the base is \( \frac{40}{2} = 20 \) miles.
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Use the Pythagorean theorem: The slant height \( l \) forms a right triangle with half the base and the height of the pyramid. Therefore, we can use the equation:
\[ l^2 = \text{height}^2 + \left(\frac{\text{base}}{2}\right)^2 \]
Plugging in the values:
\[ l^2 = 9^2 + 20^2 \]
\[ l^2 = 81 + 400 \]
\[ l^2 = 481 \]
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Calculate the slant height: Take the square root:
\[ l = \sqrt{481} \]
Approximating this gives:
\[ l \approx 21.93 \text{ miles} \]
Thus, the slant height of the square pyramid is approximately 21.93 miles.