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. 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

47.2 ft.
47.2 ft.

32.0 ft.
32.0 ft.

15.0 ft.
15.0 ft.

31.2 ft.
31.2 ft.

Answer 1

To find the height \( h \) of the square pyramid, we can use the Pythagorean Theorem. We have the following elements:

1. The slant height (\( l \)) of the pyramid is given as 25 feet.
2. The length of the front base edge is 40 feet, which means the base edge length is \( b = 40 \) feet. The distance from the center of the base to the midpoint of a side (the right base edge in this case) is \( \frac{b}{2} = \frac{40}{2} = 20 \) feet.

We can form a right triangle where:

• The height \( h \) is one leg,
• The distance from the center of the base to the midpoint of the right base edge (20 feet) is the other leg,
• The slant height (25 feet) is the hypotenuse.

According to the Pythagorean Theorem: \[ l^2 = h^2 + \left(\frac{b}{2}\right)^2 \] Substituting the known values: \[ 25^2 = h^2 + 20^2 \] Calculating the squares: \[ 625 = h^2 + 400 \] Now, isolating \( h^2 \): \[ h^2 = 625 - 400 \] \[ h^2 = 225 \] Taking the square root to find \( h \): \[ h = \sqrt{225} \] \[ h = 15 \]

So the height \( h \) of the square pyramid is \( 15.0 \) feet.

Thus, the answer is:

15.0 ft.