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.

Question

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

GPT-4o mini · Answer

To find the height $ h $ of the square pyramid, we can utilize the right triangle formed by the height, the slant height, and half the base length. First, let's denote the following: - The lateral edge length $ e = 25 $ inches - The slant height $ s = 24 $ inches - The height $ h $ (which we need to find) - The half base edge length $ \frac{b}{2} $ Since it's a square pyramid, we can use the Pythagorean theorem on the triangle formed by the height $ h $, half the base length $ \frac{b}{2} $, and the slant height $ s $: $$ s^2 = h^2 + \left(\frac{b}{2}\right)^2 $$ Unfortunately, we need to find $ b $ (the length of the base edge) before we can proceed. The relationship between the lateral edge $ e $, height $ h $, and half the base length $ \frac{b}{2} $ can also be expressed using another right triangle formed: $$ e^2 = h^2 + \left(\frac{b}{2}\right)^2 $$ Now, we have two equations: 1. $ s^2 = h^2 + \left(\frac{b}{2}\right)^2 $ 2. $ e^2 = h^2 + \left(\frac{b}{2}\right)^2 $ We can set these equal to each other and substitute $ s $ and $ e $: 1. $ 24^2 = h^2 + \left(\frac{b}{2}\right)^2 $ 2. $ 25^2 = h^2 + \left(\frac{b}{2}\right)^2 $ Calculating the squares: $$ 576 = h^2 + \left(\frac{b}{2}\right)^2 $$ $$ 625 = h^2 + \left(\frac{b}{2}\right)^2 $$ Setting both equations equal to each other, since they equal $ h^2 + \left(\frac{b}{2}\right)^2 $: $$ 576 = 625 $$ This doesn’t help as there’s no $ b $ directly relating to $ h $ and $ s $ or $ e $ yet. Instead, we'll simply begin by transforming the two equations to isolate $ h $: From $ e^2 = h^2 + \left(\frac{b}{2}\right)^2 $ and $ s^2 = h^2 + \left(\frac{b}{2}\right)^2 $, we can find that: $$ e^2 - s^2 = (h^2 + \left(\frac{b}{2}\right)^2) - (h^2 + \left(\frac{b}{2}\right)^2) $$ Rearranging gives us: $$ 625 - 576 = 49 = \left(\frac{b}{2}\right)^2 - (y^2) $$ This does not help us directly find $ h $ without $ b $. Thus we can break back from previously expressing $ \frac{b}{2} $ from both triangles' earlier approaches. Knowing: $$ \left(\frac{b}{2}\right)^2 = e^2 - h^2 = 625 - h^2 $$ And $ \left(\frac{b}{2}\right)^2 = s^2 - h^2 = 576 - h^2$ Setting these equal to each other gives: $$ 625 - h^2 = 576 - h^2 $$ The inconsistency leads us back to the equation, from the height: Using $ s = 24 \text{ and } e = 25 $ leads us to how; $ e^2 = h^2 + 576, h^2 = 25^2 - 12^2 $: $$ h^2 = 625 - 576 = 49; \ h = 7 $$ So the height $ h \approx 7 \text{ inches.} $ Thus: **Final Answer:** $ h = 7 \, \text{inches} $. (Rounded to nearest whole number.)