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.
To find the height of the pyramid, we need to find the length of r first.
Using the Pythagorean theorem, we can find the length of r:
r^2 = s^2 - e^2
r^2 = 24^2 - 25^2
r^2 = 576 - 625
r^2 = -49
Since the length cannot be negative, we discard this solution.
Therefore, we cannot find the length of h.
To find the height of the square pyramid, we will first find the length of "r" using the given slant height "s" and the lateral edge length "e".
Using the Pythagorean theorem, we can find "r" as follows:
r^2 = s^2 - e^2
r^2 = 24^2 - 25^2
r^2 = 576 - 625
r^2 = -49
Since the square of a length cannot be negative, we made an error in calculating "r". Let's try again:
r^2 = e^2 - s^2
r^2 = 25^2 - 24^2
r^2 = 625 - 576
r^2 = 49
Taking the square root of both sides gives:
r = √49
r = 7
Now that we have found the length of "r", we can use it to find the height "h". Looking at the diagram, we can see that the height "h" is perpendicular to the base, and we have a right angle where it meets the base. Therefore, we can consider the triangle formed by the height, the slant height, and "r" as a right triangle.
Using the Pythagorean theorem again, we can find "h" as follows:
h^2 = s^2 - r^2
h^2 = 24^2 - 7^2
h^2 = 576 - 49
h^2 = 527
Taking the square root of both sides gives:
h = √527
h ≈ 22.97
Rounding to the nearest whole number, the height of the square pyramid is approximately 23 inches.
To find the height of the pyramid, we first need to find the length of the dotted line labeled "r".
From the information given, we know that the slant height, labeled "s", is 24 inches. The slant height represents the distance from the vertex of the pyramid to the midpoint of one of the base edges. Since the base of the pyramid is a square, the length of each base edge is equal.
Therefore, the length of r can be found by using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides in a right triangle.
In this case, we have a right triangle formed by the slant height (hypotenuse) and the two segments labeled "r" and "s".
Using the Pythagorean theorem, we can set up the equation:
r^2 + h^2 = s^2
Plugging in the values we know:
r^2 + h^2 = 24^2
Now, we need to solve for r before finding h.
Given that the lateral edge length, labeled "e", is 25 inches, we can see that r is the hypotenuse of another right triangle formed by the height line, r, and e.
Using the Pythagorean theorem again:
r^2 = h^2 + e^2
Plugging in the values:
r^2 = h^2 + 25^2
Now we have a system of equations:
r^2 + h^2 = 24^2
r^2 = h^2 + 25^2
We can substitute the second equation into the first equation to eliminate r:
(h^2 + 25^2) + h^2 = 24^2
Simplifying:
2h^2 + 625 = 576
Subtracting 576 from both sides:
2h^2 = -49
Dividing both sides by 2:
h^2 = -24.5
Taking the square root of both sides:
h = √(-24.5)
Since the square root of a negative number is undefined and the height of a pyramid cannot be negative, it seems there may be an error in the given information or in our calculations. Please double-check the provided details and calculations to ensure accuracy.