Apply the Pythagorean theorem to determine the height of the square pyramid, h. Round the answer to the nearest tenth one side is 25 ft and the bottom is 40 ft
To apply the Pythagorean theorem, we need to identify the two legs and the hypotenuse of the right triangle formed by the height of the square pyramid.
In this case, one leg of the right triangle is the height of the pyramid, h, and another leg is half the length of the base, which is 40/2 = 20 ft.
Let's label the hypotenuse as c.
According to the Pythagorean theorem, the sum of the squares of the legs is equal to the square of the hypotenuse.
So we have: h^2 + 20^2 = c^2
Now, substitute the given values: h^2 + 400 = c^2
To isolate the height, h, we need to solve for it:
h^2 = c^2 - 400
Taking the square root of both sides, we get:
h = sqrt(c^2 - 400)
Now, we are given that one side of the square pyramid is 25 ft, which will be one of the slant heights of the triangular faces. To find c, we can use the Pythagorean theorem again:
c^2 = 25^2 + 20^2
c^2 = 625 + 400
c^2 = 1025
Taking the square root of both sides, we get:
c = sqrt(1025)
c ≈ 32.02
Now we can substitute this value of c into the equation for h:
h = sqrt(c^2 - 400)
h = sqrt((32.02)^2 - 400)
h ≈ sqrt(1024.16 - 400)
h ≈ sqrt(624.16)
h ≈ 24.97
Rounding to the nearest tenth, the height of the square pyramid, h, is approximately 25.0 ft.
To apply the Pythagorean theorem and find the height of the square pyramid, we need to understand the relationship between the height, the base, and the slant height of the pyramid.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In the case of the square pyramid, the height of the pyramid (h) is one of the legs of a right triangle, and the slant height (l) is the hypotenuse. The base of the pyramid forms the other leg of the right triangle.
Given that one side (base) is 25 ft and the bottom (another side) is 40 ft, we can find the slant height with the help of the Pythagorean theorem.
Step 1: Square the lengths of the sides:
25^2 = 625
40^2 = 1600
Step 2: Add the squares:
625 + 1600 = 2225
Step 3: Take the square root of the result to find the slant height:
√2225 ≈ 47.2 ft (rounded to the nearest tenth)
The slant height (l) of the pyramid is approximately 47.2 ft.
Now we can use the slant height and the base of the pyramid to find the height (h). Since we have a right triangle, we can use the Pythagorean theorem again:
Step 1: Square the slant height:
47.2^2 = 2227.84
Step 2: Subtract the square of the base from the above value:
2227.84 - 625 = 1602.84
Step 3: Take the square root of the result to find the height:
√1602.84 ≈ 40.1 ft (rounded to the nearest tenth)
The height (h) of the square pyramid is approximately 40.1 ft.
To determine the height of the square pyramid using the Pythagorean theorem, we can consider the right triangle formed by the height (h), half the length of the base (b/2), and one of the triangular faces.
Since one side of the square base is given as 25 ft and the bottom (another side of the base) is given as 40 ft, we can find the length of half the base by dividing 40 ft by 2:
b/2 = 40 ft / 2 = 20 ft.
Now, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, h is the hypotenuse, and b/2 and one of the triangular faces are the other two sides. So, we have:
h^2 = (b/2)^2 + (one triangular face)^2.
Plugging in the values, we get:
h^2 = 20^2 + 25^2.
Simplifying:
h^2 = 400 + 625.
h^2 = 1025.
To find the height (h), we take the square root of both sides:
h = √(1025).
Rounding the result to the nearest tenth, we get:
h ≈ 32 ft.