To find the height \( h \) of the square pyramid, we can use the relationship between the lateral edge, the slant height, and the height of the pyramid.
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Understand the dimensions: You have a square pyramid where:
- Lateral edge length \( e = 25 , \text{in} \)
- Slant height \( s = 24 , \text{in} \)
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Find the length of \( r \): The length \( r \) is half the length of one side of the base of the pyramid, which can be found using the relationship among the sides in the right triangle formed by the height \( h \), the length \( r \), and the slant height \( s \).
For a square pyramid, we have: \[ e^2 = h^2 + r^2 \] and \[ s^2 = h^2 + r^2 \]
From the given values: \[ 25^2 = h^2 + r^2 \] which simplifies to: \[ 625 = h^2 + r^2 \quad \text{(1)} \]
Also, from the slant height: \[ 24^2 = h^2 + r^2 \] which simplifies to: \[ 576 = h^2 + r^2 \quad \text{(2)} \]
Since \( s^2 \) must equal \( r^2 \), we now have two equations, which should yield the same expressions for \( h \).
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Equate the equations: From (1) and (2): \[ h^2 + r^2 = 625 \] \[ h^2 + r^2 = 576 \]
We must now analyze the steps with the correct relationships.
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Use the right triangle: Within the triangle formed by \( h \), \( r \) (where \( r \) is half of the base edge, which we can find by considering \( s \)).
Relating \( s \) and \( r \): \[ s^2 = r^2 + h^2 \rightarrow 24^2 = r^2 + h^2 \] Thus, setting up with \( h \) in terms of \( s \): \[ h^2 = s^2 - r^2 \quad \ldots \tag{3} \]
Next we can state \[ 24^2 = h^2 + r^2 \Rightarrow r^2 = 24^2 - h^2 = 576 - h^2 \]
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Substituting back into the lateral edge equation: To solve for \( h \): \[ 625 = h^2 + (576 - h^2) \] We get: \[ 625 = 576 + r^2 \rightarrow r^2 = 625 - 576 = 49 \] Then: \[ r = 7 \]
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Finding \( h \): Now, substitute back for height \( h \) using \( r^2 + h^2 = 576\): \[ h^2 = 576 - 49 \rightarrow h^2 = 527 \] Finally, solving for \( h \): \[ h = \sqrt{527} \approx 22.9 \]
Rounding to the nearest whole number gives:
The height of the square pyramid is \( 23 \) inches.