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.
To find the height of the pyramid, we need to find the length of the radius (r) first.
Knowing that the lateral edge length (e) is 25 in. and the slant height (s) is 24 in., we can use the Pythagorean theorem to find the length of r.
In a right triangle with one leg measuring e/2 and the hypotenuse measuring s, the other leg will be equal to r.
Using the Pythagorean theorem, we have:
(r)^2 + (e/2)^2 = (s)^2
(r)^2 + (25/2)^2 = (24)^2
(r)^2 + (625/4) = 576
(r)^2 = 576 - 625/4
(r)^2 = 2304/4 - 625/4
(r)^2 = 1679/4
r = √(1679/4)
Now we can find the height (h) using the Pythagorean theorem in another right triangle within the pyramid. One leg will measure r and the other will measure h, while the hypotenuse will be equal to 25 in.
(h)^2 + (r)^2 = (e)^2
(h)^2 + (√(1679/4))^2 = (25)^2
(h)^2 + 1679/4 = 625
(h)^2 = 625 - 1679/4
(h)^2 = 2500/4 - 1679/4
(h)^2 = 821/4
h = √(821/4)
Now we can calculate the value of h.
h ≈ √(821/4)
h ≈ √(205.25)
h ≈ 14.33
Rounding the height to the nearest whole number, the height of the square pyramid is approximately 14 inches.
Therefore, the height (h) of the pyramid is 14 inches.
wrong
Apologies for the error. Let's go through the calculations again.
First, we need to find the length of r. As mentioned earlier, one leg of the right triangle formed by e/2 and s is equal to r. Using the Pythagorean theorem, we have:
(r)^2 + (e/2)^2 = (s)^2
(r)^2 + (25/2)^2 = (24)^2
(r)^2 + (625/4) = 576
(r)^2 = 576 - 625/4
(r)^2 = 2304/4 - 625/4
(r)^2 = 1679/4
r = √(1679/4)
r ≈ √(419.75)
r ≈ 20.49
Now we have the length of r.
Next, we can find the height (h) using the Pythagorean theorem in another right triangle within the pyramid. One leg measures r and the other leg measures h, while the hypotenuse is equal to s.
(h)^2 + (r)^2 = (s)^2
(h)^2 + (20.49)^2 = (24)^2
(h)^2 = 576 - 419.4401
(h)^2 = 156.5599
h ≈ √156.5599
h ≈ 12.50
Rounding the height to the nearest whole number, the height of the square pyramid is approximately 13 inches.
Therefore, the height (h) of the pyramid is 13 inches.
To find the height of the pyramid, h, we first need to find the length of r.
The lateral edge length, e, is the length of one of the edges of the square base. In this case, e is given as 25 inches.
The slant height, s, is the length of one of the triangular faces of the pyramid. In this case, s is given as 24 inches.
To find the length of r, we need to use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (s) is equal to the sum of the squares of the lengths of the other two sides (r and e/2).
So, we have the equation s^2 = r^2 + (e/2)^2.
Substituting the given values, we get 24^2 = r^2 + (25/2)^2.
Simplifying, we have 576 = r^2 + (625/4).
To simplify further, we multiply everything by 4 to get rid of the fraction, which gives us 2304 = 4r^2 + 625.
Next, we subtract 625 from both sides of the equation, which gives us 2304 - 625 = 4r^2.
Simplifying, we have 1679 = 4r^2.
To isolate r, we divide both sides of the equation by 4, which gives us r^2 = 1679/4.
Taking the square root of both sides, we get r = sqrt(1679/4).
Using a calculator, we find that r is approximately 20.44 inches.
Finally, to find the height of the pyramid, h, we use the Pythagorean theorem again. This time, we have h^2 = s^2 - r^2.
Substituting the given values, we get h^2 = 24^2 - 20.44^2.
Simplifying, we have h^2 = 576 - 417.3136.
Subtracting, we get h^2 = 158.6864.
Taking the square root of both sides, we get h = sqrt(158.6864).
Using a calculator, we find that h is approximately 12.6 inches.
Rounding the answer to the nearest whole number, we get h ≈ 13 inches.
Therefore, the height of the pyramid, h, is approximately 13 inches.