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

31.2 ft.
31.2 ft.

32.0 ft.
32.0 ft.

15.0 ft.
15.0 ft.

47.2 ft.

Answer 1

To find the height \( h \) of the square pyramid using the Pythagorean theorem, we have the following information:

1. The slant height \( l \) of the pyramid is given as 25 feet.
2. The front base edge, which is a side of the base of the pyramid, is given as 40 feet.

First, we need to find the distance from the center of the base to the midpoint of one of the base edges. Since the base is square and each side is 40 feet, the distance from the center to the midpoint of one side is half of 40 feet:

\[ \text{Distance from center to midpoint} = \frac{40}{2} = 20 \text{ feet} \]

Now, we can apply the Pythagorean theorem in the right triangle formed by the height \( h \), the half base edge (20 feet), and the slant height (25 feet).

According to the Pythagorean theorem:

\[ l^2 = h^2 + \left( \frac{\text{Base Edge}}{2} \right)^2 \]

Substituting the known values:

\[ 25^2 = h^2 + 20^2 \]

Calculating the squares:

\[ 625 = h^2 + 400 \]

Now, solve for \( h^2 \):

\[ h^2 = 625 - 400 \] \[ h^2 = 225 \]

Taking the square root of both sides:

\[ h = \sqrt{225} = 15 \text{ feet} \]

Thus, the height \( h \) of the square pyramid is:

\[ \boxed{15.0 \text{ ft.}} \]