To find the height \( h \) of a square pyramid given the lateral edge length \( e \) and the slant height \( s \), we can use the following relationships:
- The slant height \( s \), the height \( h \), and the radius of the base \( r \) (which is half the side length of the base) form a right triangle:
\[
e^2 = h^2 + r^2
\]
- The slant height \( s \), the height \( h \), and the lateral edge length \( e \) are also incorporated into another right triangle formed by the vertex of the pyramid, the center of the base, and a vertex of the base:
\[
s^2 = h^2 + r^2
\]
Step 1: Find \( r \)
Since we have two equations involving \( h \):
- \( e^2 = h^2 + r^2 \)
- \( s^2 = h^2 + r^2 \)
From these, we can see both equations equal \( h^2 + r^2 \).
Let's set:
\[
e = 25 \text{ in}
\]
\[
s = 24 \text{ in}
\]
Calculating \( e^2 \) and \( s^2 \):
\[
e^2 = 25^2 = 625
\]
\[
s^2 = 24^2 = 576
\]
Step 2: Solve for \( r^2 \) from both equations
Since both \( e^2 \) and \( s^2 \) are equal to \( h^2 + r^2 \), we set them equal to each other:
\[
625 = h^2 + r^2
\]
\[
576 = h^2 + r^2
\]
From this:
\[
625 - 576 = 0 \Rightarrow 49 = h^2
\]
Step 3: Calculate \( r^2 \)
To find \( r^2 \), substitute \( h^2 = 49 \) into one of the equations:
\[
625 = 49 + r^2
\]
\[
r^2 = 625 - 49 = 576
\]
Taking the square root to find \( r \):
\[
r = \sqrt{576} = 24 \text{ in}
\]
Step 4: Find \( h \)
Now we can find the height \( h \):
\[
s^2 = h^2 + r^2
\]
\[
576 = h^2 + 576
\]
\[
h^2 = 576 - 576
\]
\[
h^2 = 0
\]
\[
h = 0 \text{ (which is incorrect, so let's check again)}
\]
Using the proper setup and solving \( h \):
Using the relationship between slant height, lateral edge, and height:
Revisiting:
Use \( e^2 = h^2 + r^2 \)
\[
625 = h^2 + 576
\]
Solving:
\[
h^2 = 625 - 576
\]
\[
h^2 = 49
\]
Hence,
\[
h = \sqrt{49} = 7 \text{ in}
\]
Result
The height \( h \) of the square pyramid is approximately 7 inches (rounded to the nearest whole number).
