To find the height \( h \) of the square pyramid, we need to first determine the radius \( r \) of the base (which is half the length of the square's diagonal), and then we can use the Pythagorean theorem in the right triangle formed by the height \( h \), the radius \( r \), and the slant height \( s \).
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Finding the radius \( r \):
Since we know the lateral edge length \( e \) (which is equal to \( 25 \) in) and the slant height \( s \) (which is equal to \( 24 \) in), we can create a right triangle that includes the slant height, the height of the pyramid, and a segment from the apex of the pyramid to the midpoint of a base edge.
In this triangle:
- The lateral edge \( e \) acts as the hypotenuse.
- The slant height \( s \) is the side opposite to the angle from the apex to the midpoint of the base edge.
- The radius \( r \) is half the length of the base edge.
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Using the Pythagorean Theorem:
In the triangle formed: \[ e^2 = h^2 + r^2 \] We also have: \[ s^2 = h^2 + r^2 \]
Using the slant height \( s \): \[ r = \sqrt{s^2 - h^2} \] We need expressions for both \( h \) and \( r \).
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Finding the height \( h \):
From the relationship \( r = \sqrt{e^2 - h^2} \) and with \( e = 25 \), we can say: \[ 25^2 = h^2 + r^2 \] \[ 625 = h^2 + r^2 \]
To express \( r \) in terms of \( s \), we also have: \[ 24^2 = h^2 + r^2 \] \[ 576 = h^2 + r^2 \]
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Setting equations for \( h \):
We have two equations now, and we can solve these to find \( h \): \[ 625 = h^2 + r^2 \quad (1) \] \[ 576 = h^2 + r^2 \quad (2) \]
This implies: \[ h^2 + r^2 = 625 \] \[ h^2 + r^2 = 576 \]
From equation (1) and (2), we now focus on using the lateral edge: \[ e = \sqrt{h^2 + r^2} \] Using the Law of Cosines (where \( L = 25 \)): \[ 25^2 = h^2 + [(24^2 - h^2)] \] \[ 645 - (576 - h^2) = 625 \] Then plug in: \[ h^2 = 625 - r^2 \]
To find numerical values, we need to start systematically to isolate \( h \).
If we assume \( r \) is equal to multiple equations in our calculations, but using \( (e^2 - s^2) = r^2 \): Taking trial errors: \[ e^2 = r^2 + h^2 \rightarrow h = e^2 - s^2 = \sqrt{625 - 576} = \sqrt{49} = 7. \]
Thus, calculate to find ( r: r^2 = 625 - (7^2) \rightarrow 625 - 49 = 576 ] Thus, we find: \[ h = \sqrt{576 - 24^2} \]
Finally, we compute the height \( h \) = 7 in, directly rounded from our initial trial error relationships.
Therefore, the height of the square pyramid is \( 7 \) in.