To find the height \( h \) of the square pyramid with given lateral edge length \( e = 25 \) inches and slant height \( s = 24 \) inches, we first need to determine the length of \( r \), which is half the length of the base side of the pyramid.
In a square pyramid, the relationship between the slant height \( s \), the height \( h \), and half the base side \( r \) can be expressed using the Pythagorean theorem:
\[ s^2 = h^2 + r^2 \]
From the problem, we also know the relationship between the lateral edge \( e \), the height \( h \), and half the diagonal of the base. The half diagonal \( d \) of the base can be calculated from the side length of the base, which is \( 2r \):
\[ d = \sqrt{(2r)^2 + (2r)^2} = \sqrt{8r^2} = 2\sqrt{2}r \]
The relationship involving the lateral edge \( e \) can also be expressed using the Pythagorean theorem:
\[ e^2 = h^2 + d^2 \]
Substituting for \( d \):
\[ e^2 = h^2 + (2\sqrt{2}r)^2 = h^2 + 8r^2 \]
Now we have two equations:
- \( s^2 = h^2 + r^2 \)
- \( e^2 = h^2 + 8r^2 \)
Now, let's substitute the known values:
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For \( s = 24 \): \[ 24^2 = h^2 + r^2 \implies 576 = h^2 + r^2 \quad (1) \]
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For \( e = 25 \): \[ 25^2 = h^2 + 8r^2 \implies 625 = h^2 + 8r^2 \quad (2) \]
Now we can solve these two equations simultaneously. From equation (1), we can express \( h^2 \):
\[ h^2 = 576 - r^2 \]
Now, substitute \( h^2 \) into equation (2):
\[ 625 = (576 - r^2) + 8r^2 \] \[ 625 = 576 + 7r^2 \] \[ 625 - 576 = 7r^2 \] \[ 49 = 7r^2 \implies r^2 = 7 \implies r = \sqrt{7} \]
Now substituting \( r^2 = 7 \) back into equation (1) to find \( h^2 \):
\[ h^2 = 576 - 7 \] \[ h^2 = 569 \] \[ h = \sqrt{569} \approx 23.8 \]
Rounding to the nearest whole number, the height \( h \) of the square pyramid is:
\[ \boxed{24} \text{ inches} \]