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. The back base edge is not visible but its position is indicated by a dashed line. The front base edge shows a length of 40 feet. 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. 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. This dotted line shows a slant height of 25 feet.

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. The back base edge is not visible but its position is indicated by a dashed line. The front base edge shows a length of 40 feet. 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. 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. This dotted line shows a slant height of 25 feet.
Apply the Pythagorean Theorem to determine the height of the square pyramid,  h
 . Round the answer to the nearest tenth.

(1 point)
Responses

32.0 ft. 
32.0 ft.

31.2 ft. 
31.2 ft.

47.2 ft.
47.2 ft.

15.0 ft.

GPT-4o mini · Answer

To find the height $ h $ of the square pyramid using the Pythagorean Theorem, we need to consider the right triangle formed by the height $ h $, half the length of the base edge, and the slant height. 1. **Identify the elements of the triangle**: - The length of the base edge is given as 40 feet, so half of that (the distance from the center of the base to the edge of the base) is: $$ \frac{40}{2} = 20 \text{ feet} $$ - The slant height $ l $ is given as 25 feet. 2. **Set up the Pythagorean Theorem**: We use the Pythagorean theorem in the context of the right triangle formed: $$ h^2 + 20^2 = 25^2 $$ 3. **Calculate $ 20^2 $ and $ 25^2 $**: $$ 20^2 = 400 $$ $$ 25^2 = 625 $$ 4. **Substitute these values into the equation**: $$ h^2 + 400 = 625 $$ 5. **Solve for $ h^2 $**: $$ h^2 = 625 - 400 $$ $$ h^2 = 225 $$ 6. **Take the square root of both sides to find $ h $**: $$ h = \sqrt{225} = 15 \text{ feet} $$ Rounding to the nearest tenth, the height $ h $ of the pyramid is $ 15.0 $ ft. Thus, the correct response is: **15.0 ft.**