Use the image to answer the question.

Question

An illustration shows a square pyramid with labeled parts. The left, front, and right faces of the pyramid and base are visible. The back face is not visible, but its position is indicated by a dashed line. The back base edge is not visible, but its position is indicated by a dashed line. The left base edge is labeled a. The back left lateral edge is labeled e. A dotted line extends from the interior of the vertex straight down to the center of the base and is labeled h. A right angle is indicated where the dotted height line meets the base. Another dotted line extends from the center of the base where the height line meets to the center of the right base edge and is labeled r. A dotted line extends from the right base edge straight up the center of the right face of the prism and meets the height line at the vertex. The dotted slant height line is labeled s.

For the square pyramid, the lateral edge length, e
e
, is 25 in., and the slant height, s
s
, is 24 in. Find the height of the pyramid, h
h
. Round the answer to the nearest whole number. Hint: Before finding the length of h
h
, you will first need to find the length of r
r
.

Answer 1

To find the height of the pyramid, \(h\), we first need to find the length of \(r\), which is the distance from the center of the base to the midpoint of one of the base edges.

We know that in a square pyramid, the lateral edge length \(e\) and the slant height \(s\) are related to the height \(h\) and the half base length \(r\) in the following way:

From the right triangle formed by the height \(h\), \(r\), and the slant height \(s\), we can apply the Pythagorean theorem:

\[ s^2 = h^2 + r^2 \]

Given:

• Lateral edge length, \(e = 25\) inches
• Slant height, \(s = 24\) inches

Next, we also need to establish a relationship between \(e\), \(h\), and \(r\). In another right triangle formed by the lateral edge \(e\), height \(h\), and the distance \(r\) (half of the base edge) from the midpoint of the base edge to the apex, we set up another equation:

\[ e^2 = h^2 + r^2 \]

Now substitute \(s\) into the first equation:

\[ 24^2 = h^2 + r^2 \]

Calculating \(24^2\):

\[ 576 = h^2 + r^2 \quad (1) \]

Next, we use the second equation:

\[ 25^2 = h^2 + r^2 \]

Calculating \(25^2\):

\[ 625 = h^2 + r^2 \quad (2) \]

Now, we have two equations, (1) and (2):

From (1): \[ h^2 + r^2 = 576 \]

From (2): \[ h^2 + r^2 = 625 \]

Now let's isolate \(r^2\) from both equations.

Let's subtract equation (1) from equation (2):

\[ 625 - 576 = 0 \]

This results in:

\[ 49 = r^2 \]

Thus, to find \(r\):

\[ r = \sqrt{49} = 7 \text{ inches} \]

Next, we can use the value of \(r\) to substitute back into either equation to find \(h\). Using equation (1):

\[ h^2 + 7^2 = 576 \] \[ h^2 + 49 = 576 \] \[ h^2 = 576 - 49 \] \[ h^2 = 527 \] \[ h = \sqrt{527} \approx 22.9 \]

Rounding to the nearest whole number, the height \(h\) is:

\[ \boxed{23} \text{ inches} \]