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
15.0 ft.
15.0 ft.
32.0 ft.
32.0 ft.
47.2 ft.
47.2 ft.
31.2 ft.
31.2 ft.
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To determine the height of the square pyramid, we can use the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).
In this case, we can consider the height line (h) as the hypotenuse. The dotted line extending from the right base edge to the center of the right face of the prism can be considered as one leg, and the slant height of 25 feet can be considered as the other leg.
Let's label the leg from the right base edge to the center of the right face as a, and the slant height as b.
Using the Pythagorean Theorem, we have:
a^2 + b^2 = h^2
Given that the slant height is 25 feet, we can substitute b = 25 into the equation:
a^2 + 25^2 = h^2
Now, we need to determine the length of a. We can use the fact that the front base edge is 40 feet, so the length of half of the front base edge is 20 feet (since it is a square pyramid).
Since the dotted line extends from the center of the base where the height line meets to the center of the right base edge, it forms a right angle. Thus, we can consider the line from the center of the base to the right base edge as the hypotenuse of a right triangle, and half of the front base edge (20 feet) as one leg. Let's label the length of this line as c.
Using the Pythagorean Theorem again, we have:
20^2 + c^2 = a^2
Simplifying this equation, we have:
400 + c^2 = a^2
Since the length of the left, front, and right faces of the pyramid are equal, the length of the line from the center of the base to the right base edge (c) is the same as the length of half of the front base edge, which is 20 feet.
Substituting c = 20 into the equation, we have:
400 + 20^2 = a^2
400 + 400 = a^2
800 = a^2
a = √800 ≈ 28.3
Now, substituting the values of a and b into the equation a^2 + 25^2 = h^2, we have:
(28.3)^2 + 25^2 = h^2
800 + 625 = h^2
1425 = h^2
Taking the square root of both sides, we have:
h = √1425 ≈ 37.8
Rounding to the nearest tenth, the height of the square pyramid is approximately 37.8 feet.