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

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
 , 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 need to find the length of  r
 .

GPT-4o mini · Answer

To find the height $ h $ of the square pyramid, we first need to find $ r $, the distance from the center of the base to the midpoint of one of the base edges. Since the base of the pyramid is square, we can denote the length of one side of the base as $ a $. From the information given: - The lateral edge length $ e $ is 25 in. - The slant height $ s $ is 24 in. The relationship between the slant height $ s $, the height $ h $, and the distance $ r $ from the center of the base to the midpoint of a base edge can be illustrated using the right triangle formed by $ h $, $ r $, and $ s $. 1. **Finding $ r $**: The distance $ r $ is half the length of one side of the base $ a $: $$ r = \frac{a}{2} $$ 2. **Using the right triangle**: In the right triangle formed: $$ s^2 = h^2 + r^2 $$ Substituting $ s = 24 $: $$ 24^2 = h^2 + r^2 $$ $$ 576 = h^2 + r^2 $$ 3. **Finding $ a $**: To find $ a $, we also use the relationship involving the lateral edge $ e $: The lateral edge $ e $ and the height $ h $ relate to $ r $: $$ e^2 = h^2 + r^2 $$ Substituting $ e = 25 $, we get: $$ 25^2 = h^2 + r^2 $$ $$ 625 = h^2 + r^2 $$ 4. **Setting up the equations**: Now we have two equations: $$ 576 = h^2 + r^2 \quad \text{(1)} $$ $$ 625 = h^2 + r^2 \quad \text{(2)} $$ From equation (1), we can rearrange to express $ h^2 $: $$ h^2 = 576 - r^2 $$ Substitute this into equation (2): $$ 625 = (576 - r^2) + r^2 $$ This simplifies to: $$ 625 = 576 $$ This indicates that there is a misunderstanding in how these relationships are being defined. We should properly rethink how $ r $ affects the structures of triangles. For the case and working through the geometry: We first look at the lateral triangle where: $$ h^2 + r^2 = s^2 $$ Knowing that $ s = 24 $ we can express height directly as we calculate edge dimensions accordingly from the base calculations separately. 5. **Digging back for basic values**: By finding $ r$ from the dimensional ratios of square pyramid dimensions directly; Geometry ensures $\frac{25^2 - 24^2}{2} = h$. 6. **Final calculation**: Use: $$ r^2 = s^2 - h^2 $$ $$ 24^2 = h^2 + r^2; $$ This associates: $$ h^2 = 25^2 - r^2}, $$ To get directly: $$ I find h = \sqrt{25*25-24*24} ≈ 7 $$ Make adjustments accordingly with specific changes in measurement ideally around the square. Solve geometrically or via functional square pyramid property orthogonal bases. **So, height:** $$ h \approx 7.0\; in. $$ Thus round to $ 7 \; in$ as height.