To find the height \( h \) of the square pyramid, we can utilize the relationships among the lateral edge length \( e \), the slant height \( s \), and the height \( h \) of the pyramid.
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Find the length of the radius \( r \): The radius \( r \) of the base of a square pyramid is half of the length of one side of the base, which can also be expressed using the relationships in a right triangle. The relationship between the lateral edge \( e \), the height \( h \), and the radius \( r \) can be expressed as: \[ e^2 = h^2 + r^2 \]
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We can also express the relationship between the slant height \( s \), the height \( h \), and the radius \( r \): \[ s^2 = h^2 + r^2 \]
Now we can set both equations together: \[ e^2 = h^2 + r^2 \quad \text{(1)} \] \[ s^2 = h^2 + r^2 \quad \text{(2)} \]
Since both equations equal \( h^2 + r^2 \), we can equate them: \[ e^2 = s^2 \] Substituting the values: \[ 25^2 = 24^2 + r^2 \] \[ 625 = 576 + r^2 \] \[ r^2 = 625 - 576 = 49 \] \[ r = \sqrt{49} = 7 \text{ in.} \]
- Finding the height \( h \): Now, we can use equation (1) to find \( h \): \[ 25^2 = h^2 + 7^2 \] \[ 625 = h^2 + 49 \] \[ h^2 = 625 - 49 = 576 \] \[ h = \sqrt{576} = 24 \text{ in.} \]
Thus, the height of the square pyramid is approximately \( \boxed{24} \) in.
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