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?
To solve this problem, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the height of the cone is one of the sides of a right-angled triangle, and the slant height is the hypotenuse.
We know that the radius of the cone is 10, and the height is 13.
Using the Pythagorean theorem, we can calculate the length of the slant height:
l^2 = h^2 + r^2
l^2 = 13^2 + 10^2
l^2 = 169 + 100
l^2 = 269
Taking the square root of both sides, we find:
l = sqrt(269)
Therefore, the length of the slant height is approximately sqrt(269) feet.
Since the height of the cone is the other side of the right-angled triangle, we can use the Pythagorean theorem again to find the length of the height:
h^2 = l^2 - r^2
h^2 = (sqrt(269))^2 - 10^2
h^2 = 269 - 100
h^2 = 169
Taking the square root of both sides, we find:
h = sqrt(169)
Therefore, the length of the height of the cone is approximately sqrt(169) = 13 feet.