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

32.0 ft.
32.0 ft.

15.0 ft.
15.0 ft.

47.2 ft.
47.2 ft.

31.2 ft.

Answer 1

To find the height \( h \) of the square pyramid using the Pythagorean theorem, we will set up the problem based on the information given.

We have:

• The slant height \( l = 25 \) ft.
• The half of the front base edge, which is half of \( 40 \) ft, so \( \frac{40}{2} = 20 \) ft.

In the right triangle formed by the height \( h \), half the base length \( 20 \) ft, and the slant height \( 25 \) ft, we can apply the Pythagorean theorem:

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

Substituting the values we have:

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

Calculating the squares:

\[ 625 = h^2 + 400 \]

Now, isolate \( h^2 \):

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

Taking the square root:

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

So the height of the square pyramid is \( \boxed{15.0 \text{ ft}} \).