Use the image to answer the 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.
To apply the Pythagorean Theorem in this case, we can consider the right triangle formed by the slant height, the height, and the line from the center of the base to the center of the right base edge.
Let's label the slant height as s, the height as h, and the length from the center of the base to the center of the right base edge as c.
Using the Pythagorean Theorem, we have:
s^2 = h^2 + c^2
Given that the slant height is 25 feet, we can write:
25^2 = h^2 + c^2
625 = h^2 + c^2
Now, looking at the right triangle formed by the height, the line from the center of the base to the center of the right base edge, and the back base edge, we have another right triangle.
The length from the center of the base to the center of the right base edge is given as 40 feet, which is equal to c. The length from the center of the base to the back base edge, which is not visible in the image, can be determined using the fact that it is perpendicular to the front base edge. Since the front base edge has a length of 40 feet, the back base edge will also have a length of 40 feet.
Let's label the length from the center of the base to the back base edge as a. Now, we can also apply the Pythagorean Theorem to find the value of a:
a^2 = h^2 + 40^2
a^2 = h^2 + 1600
Now, let's combine the equation for a^2 with the previous equation for s^2:
625 = h^2 + c^2
a^2 = h^2 + 1600
We can substitute the value of a^2 from the second equation into the first equation:
625 = (h^2 + 1600) + c^2
Rearranging the equation, we have:
h^2 + c^2 = 625 - 1600
h^2 + c^2 = -975
Since the Pythagorean Theorem cannot yield a negative result, it seems there may be an error in the given measurements or description of the image. Without further information or correction, it is not possible to determine the height of the square pyramid using the Pythagorean Theorem. Thus, none of the response options provided are correct.