SOLUTION: The diameter of the base of a right circular cone is 10 in., and its altitude is 8 in. Find the volume of the largest sphere that can be cut from the cone.

Algebra ->  Volume -> SOLUTION: The diameter of the base of a right circular cone is 10 in., and its altitude is 8 in. Find the volume of the largest sphere that can be cut from the cone.      Log On


   

Question 1181733: The diameter of the base of a right circular cone is 10 in., and its altitude is 8 in. Find the volume of the largest sphere that can be cut from the cone.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


In cross section, the diagram will be a circle inscribed in an isosceles triangle with base 10 and height 8; the center of the sphere (circle in the cross section) is the same distance from all three sides of the triangle.

Draw the figure, showing the altitude of the triangle and the three radii from the center of the circle to the three sides of the triangle.

In your diagram there is a right triangle with short leg length 5, long leg 8, and hypotenuse sqrt(89).

And there is a second right triangle, similar to the first because of equal angles, that has short leg equal to the radius of the circle and hypotenuse equal to 8 minus the radius of the circle.

Find the radius of the circle (sphere) using the known lengths of the short leg and hypotenuse of those two similar triangles.

%288-r%29%2Fr+=+sqrt%2889%29%2F5
r%2Asqrt%2889%29=5%288-r%29
r%2Asqrt%2889%29+=+40-5r
r%28sqrt%2889%29%2B5%29=40
r+=+40%2F%28sqrt%2889%29%2B5%29

Do that calculation to find the radius of the circle(sphere); then use that radius in the formula for the volume of a sphere to find the answer to the problem.

-----------------------------------------------------------

Note my answer to 2 decimal places is 89.15