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, but its position is indicated by a dashed line. The back base edge is not visible, but its position is indicated by a dashed line. The left base edge is labeled a. The back left lateral edge is labeled e. 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 and is labeled r. 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. The dotted slant height line is labeled s.
For the square pyramid, the lateral edge length, e , is 25 in., and the slant height, s , is 24 in. Find the height of the pyramid, h . Round the answer to the nearest whole number. Hint: Before finding the length of h , you will first need to find the length of r
.
(1 point)
The height of the square pyramid is
in.
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To find the length of r, we can use the Pythagorean theorem in the right triangle formed by r, h, and s.
r^2 + h^2 = s^2
r^2 + h^2 = 24^2
r^2 + h^2 = 576
We also know that r is the hypotenuse of the right triangle formed by a, h, and r, so:
a^2 + h^2 = r^2
a^2 + h^2 = 25^2
a^2 + h^2 = 625
Now we can substitute r^2 from the first equation into the second equation:
a^2 + h^2 = 625
a^2 + 576 - h^2 = 625
a^2 = 49
a = 7 in.
Now we can find h using the Pythagorean theorem:
7^2 + h^2 = 576
49 + h^2 = 576
h^2 = 527
h ≈ 23 in.
Therefore, the height of the square pyramid is approximately 23 inches.