Observe the cone figure below. The slant height, l, is the diagonal length of the cone. The cone stands 13 feet tall and has a radius of 10, what is the length of the height of the cone?
a2 +b2 = c2
length of the height equals?
Using the Pythagorean theorem, we have:
l^2 = h^2 + r^2
Since the radius is given to be 10, we can substitute this value into the equation:
l^2 = h^2 + 10^2
l^2 = h^2 + 100
Given that the cone is 13 feet tall, we can also substitute this value into the equation:
l^2 = 13^2 + 100
l^2 = 169 + 100
l^2 = 269
Taking the square root of both sides, we have:
l ≈ √269
So, the approximate length of the slant height is √269 feet.
i got the same answer but it says it isnt right!
I apologize for the confusion. Let's go through the calculation again.
Using the Pythagorean theorem, we have:
l^2 = h^2 + r^2
Given that the radius is 10 and the cone is 13 feet tall, we can substitute these values into the equation:
l^2 = h^2 + 10^2
l^2 = h^2 + 100
Now, let's solve for h. Rearrange the equation:
h^2 = l^2 - 100
Substituting the value of l^2 = 13^2 = 169:
h^2 = 169 - 100
h^2 = 69
Taking the square root of both sides, we have:
h ≈ √69
So, the approximate length of the height of the cone is √69 feet.
To find the length of the height of the cone, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, we can consider the slant height of the cone (l) as the hypotenuse, the height of the cone (h) as one of the legs, and the radius of the base (r) as the other leg. The cone can be visualized as a right triangle, with the slant height as the diagonal, the height as the vertical side, and the radius as the horizontal side.
According to the Pythagorean theorem:
l^2 = h^2 + r^2
Since we are given the slant height (l) as the diagonal, the height (h) as 13 feet, and the radius (r) as 10 feet, we can substitute these values into the equation.
l^2 = h^2 + r^2
l^2 = 13^2 + 10^2
l^2 = 169 + 100
l^2 = 269
To find the length of the height (h), we need to isolate h in the equation. Taking the square root of both sides of the equation:
√(l^2) = √(h^2 + r^2)
l = √(h^2 + r^2)
Applying this to our equation:
√269 = √(h^2 + 10^2)
√269 = √(h^2 + 100)
Now we can solve for h. To do this, we square both sides of the equation:
269 = h^2 + 100
Rearranging the equation, we get:
h^2 = 269 - 100
h^2 = 169
Taking the square root of both sides again:
√(h^2) = √169
h = 13
Therefore, the length of the height of the cone is 13 feet.