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
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To find the height of the square pyramid, we need to find the length of the radius (r) first.
In a square pyramid, the slant height (s) is the hypotenuse of a right triangle formed by the height (h) and the lateral edge length (e).
We can use the Pythagorean theorem to find the length of the radius:
r^2 = s^2 - (e/2)^2
Let's substitute the values given:
r^2 = 24^2 - (25/2)^2
Now we can simplify the equation:
r^2 = 576 - (625/4)
To subtract fractions, we need a common denominator. The common denominator for 1 and 4 is 4.
r^2 = 576 - (625/4)(4/4)
r^2 = 576 - 2500/4
To simplify further, we need to convert the whole number 576 into a fraction with a denominator of 4.
r^2 = (2304/4) - (2500/4)
r^2 = (-196/4)
Now we can calculate the square root of both sides:
r = sqrt(-196/4)
Since we're looking for a positive length, we can disregard the negative square root.
r = sqrt(196/4)
r = sqrt(49)
r = 7
Therefore, the length of the radius (r) is 7 inches.
Now that we know the length of the radius, we can find the height (h) by using the Pythagorean theorem again:
h^2 = s^2 - r^2
h^2 = 24^2 - 7^2
h^2 = 576 - 49
h^2 = 527
To find the height, we take the square root of both sides:
h = sqrt(527)
h ≈ 22.94
Rounding the answer to the nearest whole number, the height (h) of the square pyramid is 23 inches.
23 in.
To find the height of the pyramid, h, we first need to find the length of the radius, r.
Let's start by using the given slant height, s, and the lateral edge length, e, to find the length of the radius, r.
In a square pyramid, the slant height, s, and the lateral edge length, e, form a right triangle with the height, h, of the pyramid.
Using the Pythagorean theorem, we can find the length of the radius, r:
r^2 = s^2 - (e/2)^2
Substituting the given values:
r^2 = 24^2 - (25/2)^2
r^2 = 576 - (625/4)
r^2 = 576 - 156.25
r^2 = 419.75
Taking the square root of both sides to solve for r:
r ≈ √419.75
r ≈ 20.49
Now that we have found the length of the radius, we can use it to find the height, h, of the pyramid.
In a square pyramid, the height, h, can be found using the equation:
h = √(s^2 - r^2)
Substituting the given value of the slant height, s, and the length of the radius, r:
h = √(24^2 - 20.49^2)
h = √(576 - 419.75)
h = √(156.25)
h ≈ 12.5
Therefore, the height of the pyramid, h, rounded to the nearest whole number, is 13 inches.
To find the height of the pyramid, we first need to find the length of the radius of the base.
The radius of the base of a square pyramid is equal to half of the length of one side of the base. Since the lateral edge length is 25 inches, the length of one side of the base is also 25 inches. Therefore, the radius of the base is 25/2 = 12.5 inches.
Now, we can use the Pythagorean theorem to find the height of the pyramid. The height, slant height, and radius form a right triangle. The slant height is the hypotenuse of this triangle, the height is one of the legs, and the radius is the other leg.
Using the Pythagorean theorem, we have:
h^2 + r^2 = s^2
Substituting the known values, we get:
h^2 + 12.5^2 = 24^2
h^2 + 156.25 = 576
h^2 = 576 - 156.25
h^2 = 419.75
Taking the square root of both sides, we find:
h = √419.75
h ≈ 20.5
Therefore, the height of the pyramid is approximately 20.5 inches, rounding to the nearest whole number, it is 21 inches.