To determine the length of the base of a square pyramid given the height from the base to the apex and the slant height, we can use the following method:
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Understand the dimensions:
- Let \( h \) be the height of the pyramid (20 m from the base to the apex).
- Let \( l \) be the slant height (25 m).
- Let \( a \) be the length of the base of the square pyramid.
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Use the relationship: In a right triangle formed by the height (\( h \)), half the base length (\( \frac{a}{2} \)), and the slant height (\( l \)), we can apply the Pythagorean theorem:
\[ l^2 = h^2 + \left(\frac{a}{2}\right)^2 \]
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Substitute the known values:
- \( l = 25 , \text{m} \)
- \( h = 20 , \text{m} \)
Plugging in the values, we get:
\[ 25^2 = 20^2 + \left(\frac{a}{2}\right)^2 \]
Thus:
\[ 625 = 400 + \left(\frac{a}{2}\right)^2 \]
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Solve for \( \frac{a}{2} \):
\[ 625 - 400 = \left(\frac{a}{2}\right)^2 \]
\[ 225 = \left(\frac{a}{2}\right)^2 \]
Taking the square root of both sides gives:
\[ \frac{a}{2} = 15 \]
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Calculate the base length \( a \):
\[ a = 2 \times 15 = 30 , \text{m} \]
Therefore, the length of the base of the square pyramid is 30 meters.
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