To find the height \( h \) of the square pyramid, we will first determine the length of \( r \), which is the distance from the center of the base to the midpoint of the base edge.
Given that the lateral edge length \( e = 25 \) in and the slant height \( s = 24 \) in, we can visualize a right triangle formed by the height \( h \), the slant height \( s \), and the distance \( r \).
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Since the pyramid's base is a square, the distance \( r \) (half the length of one side of the base) can be calculated using the Pythagorean theorem: \[ e^2 = h^2 + r^2 \] We can express \( r \) in terms of the edge length of the base \( a \) as: \[ r = \frac{a}{2} \]
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Now we know \( s^2 = h^2 + r^2 \). So we can express \( r \) in terms of \( s \) and \( h \): \[ r = \sqrt{s^2 - h^2} \]
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Since we need \( r \) in terms of \( h \) to solve for \( h \) eventually, we notice that from the information available, we don't have the side length \( a \), but we can calculate \( r \) using a derived relationship.
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Since we are given that: \[ e^2 = h^2 + r^2 \tag{1} \] And also since we earlier stated that: \[ r = \sqrt{s^2 - h^2} \tag{2} \]
Setting up these equations, we can formulate it based on \( e = 25 \) and \( s = 24 \):
Using equation (1): \[ 25^2 = h^2 + r^2 \implies 625 = h^2 + r^2 \] Also using equation (2): \[ r^2 = s^2 - h^2 \implies r^2 = 24^2 - h^2 \implies r^2 = 576 - h^2 \]
Substituting \( r^2 \) into the equation we have: \[ 625 = h^2 + (576 - h^2) \] This will simplify: \[ 625 = 576 + h^2 \implies h^2 = 625 - 576 = 49 \implies h = \sqrt{49} = 7 \]
Thus, the height of the square pyramid \( h \) is: \[ \boxed{7} \text{ in.} \]