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.
The height of the square pyramid is in
To find the height of the pyramid, h, we first need to find the length of the radius, r.
Since a square pyramid has a square base, the length of each side of the base is equal to e (25 in).
The slant height, s, is the hypotenuse of the right triangle formed by the height, h, the radius, r, and the slant height.
Using the Pythagorean theorem, we can write s^2 = h^2 + r^2.
Substituting the given values, we have (24 in)^2 = h^2 + r^2.
Expanding, we get 576 in^2 = h^2 + r^2.
Since the base of the pyramid is a square, the length of each side is equal to e.
Using Pythagorean theorem again on the base, we have e^2 = r^2 + r^2.
Expanding, we get 25^2 = 2r^2.
Simplifying, we have 625 = 2r^2.
Dividing both sides by 2, we get 312.5 = r^2.
Taking the square root of both sides, we get r ≈ 17.68.
Now that we know the value of r, we can substitute it back into the first equation.
576 = h^2 + (17.68)^2.
Simplifying, we get 576 = h^2 + 312.5.
Subtracting 312.5 from both sides, we get 263.5 = h^2.
Taking the square root of both sides, we get h ≈ 16.23.
Rounding to the nearest whole number, the height of the square pyramid is approximately 16 inches.
That is incorrect try again
Apologies for the mistake. Let's try again to find the height of the pyramid, h.
To find the height, we can use the Pythagorean theorem. In a right triangle formed by the height, h, the radius, r, and the slant height, s, we have:
s^2 = h^2 + r^2
Substituting the given values, we have:
(24 in)^2 = h^2 + r^2
Expanding, we get:
576 in^2 = h^2 + r^2
Since the base of the pyramid is a square, the length of each side is equal to the lateral edge length, e:
e = r
Substituting the given value, we have:
25 in = r
Now we can substitute r = 25 in in the equation:
576 in^2 = h^2 + (25 in)^2
Expanding, we get:
576 in^2 = h^2 + 625 in^2
Subtracting 625 in^2 from both sides, we get:
576 in^2 - 625 in^2 = h^2
-49 in^2 = h^2
Taking the square root of both sides, we get:
h ≈ √(-49 in^2)
Since we can't take the square root of a negative number with real numbers, this means that there is no real solution for the height of the pyramid.
Try doing it in the The Pythagorean Theorem in Three Dimensions
Apologies for the confusion. Let's use the Pythagorean theorem in three dimensions to find the height of the pyramid, h.
In a square pyramid, the height, h, slant height, s, and lateral edge length, e, form a right triangle. Therefore, we can use the Pythagorean theorem in three dimensions:
h^2 = s^2 - (e/2)^2
Substituting the given values, we have:
h^2 = (24 in)^2 - (25 in / 2)^2
Calculating the values, we get:
h^2 = 576 in^2 - (625 in^2 / 4)
Simplifying, we have:
h^2 = 576 in^2 - 156.25 in^2
h^2 = 419.75 in^2
Calculating the square root of both sides, we get:
h ≈ √(419.75 in^2)
h ≈ 20.49 in
Rounding to the nearest whole number, the height of the square pyramid is approximately 20 inches.
A juice can is in the shape of a cylinder. It has a height of 122 mm and a diameter of 52 mm. What is the longest straw that can fit completely inside the juice can diagonally? Round the answer to the nearest tenth. (1 point)
To find the longest straw that can fit completely inside the juice can diagonally, we need to find the diagonal of the cylinder.
The diameter of the cylinder is 52 mm, which means the radius, r, is half the diameter or 26 mm.
The height of the cylinder is 122 mm.
The diagonal can be found using the Pythagorean theorem. In a right triangle formed by the height, the radius, and the diagonal, we have:
diagonal^2 = height^2 + radius^2
Substituting the given values, we have:
diagonal^2 = (122 mm)^2 + (26 mm)^2
Calculating the values, we get:
diagonal^2 = 14884 mm^2 + 676 mm^2
diagonal^2 = 15560 mm^2
Calculating the square root of both sides, we get:
diagonal ≈ √(15560 mm^2)
diagonal ≈ 124.8 mm
Rounding to the nearest tenth, the longest straw that can fit completely inside the juice can diagonally would be approximately 124.8 mm.