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

15.0 ft.

32.0 ft.

47.2 ft.

31.2 ft.

Answer 1

To find the height (h) of the square pyramid using the Pythagorean theorem, we can analyze the right triangle formed by the height (h), half the base length (which is 20 feet), and the slant height (25 feet).

The base length of the pyramid is given as 40 feet. Thus, half of the base length is:

\[ \frac{40}{2} = 20 , \text{feet} \]

Now, we have:

• One leg of the right triangle: half the base = 20 feet
• The other leg (the height): \( h \)
• The hypotenuse: slant height = 25 feet

According to the Pythagorean theorem:

\[ a^2 + b^2 = c^2 \]

Here, let \( a = h \), \( b = 20 \), and \( c = 25 \).

Substituting the values in gives:

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

Calculating \( 20^2 \) and \( 25^2 \):

\[ h^2 + 400 = 625 \]

Now, we can isolate \( h^2 \):

\[ h^2 = 625 - 400 \]

\[ h^2 = 225 \]

Taking the square root of both sides:

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

Thus, the height \( h \) is:

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