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.

31.2 ft.
31.2 ft.

15.0 ft.
15.0 ft.
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Answer 1

To find the height \( h \) of the square pyramid using the Pythagorean Theorem, we can set up the relationship between the height, half the base edge length, and the slant height.

From the illustration described, we know:

• The full length of the base edge is \( 40 \) feet.
• Therefore, half the base edge length is \( \frac{40}{2} = 20 \) feet.
• The slant height \( s \) is given as \( 25 \) feet.

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the slant height, in this case) is equal to the sum of the squares of the other two sides (the height and half the base edge):

\[ s^2 = h^2 + \left(\frac{b}{2}\right)^2 \]

Substituting the known values:

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

Calculating the squares:

\[ 625 = h^2 + 400 \]

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

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

Now, take the square root of both sides to find \( h \):

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

Thus, the height of the pyramid \( h \) is \( 15.0 \) ft.

The correct response is:

15.0 ft.