Questions on Geometry: Bodies in space, right solid, cylinder, sphere answered by real tutors!

Algebra ->  Algebra -> Questions on Geometry: Bodies in space, right solid, cylinder, sphere answered by real tutors!     (Log On)
Ad: Algebra Solved!™: algebra software that solves YOUR algebra homework problems with step-by-step help!


Question 91638: A solid wooden cylinder has a height equal to the diameter of the base circle. This cylinder is then shaved until a sphere is created. If the resultant sphere is the largest possible sphere, determine what proportion of the original cylinder must be discarded in order to produce this sphere. Show working to justify your answer.: A solid wooden cylinder has a height equal to the diameter of the base circle. This cylinder is then shaved until a sphere is created. If the resultant sphere is the largest possible sphere, determine what proportion of the original cylinder must be discarded in order to produce this sphere. Show working to justify your answer.
Answer by Earlsdon(3816) About Me  (Show Source):
You can put this solution on YOUR website!
Let's start with the appropriate formulas for the volumes of the given solids.
Cylinder:
V[c] = (pi)r^2h but the height of the cylinder is given as the length of the diameter (D) of its base and D = 2r where r is the radius of the base. So we can express the volume of the cylinder entirely in terms of the radius (r) of its base, right?
V[c] = (pi)r^2(2r) Simplifying this we get:
V[c] = 2(pi)r^3 as the volume of the cylinder.
Sphere:
The largest sphere that could be contained within a cylinder of the dimensions given above i.e., h = 2r and radius r, would be a sphere whose radius is equal to that of the cylinder, or radius r.
The volume of a sphere of radius r is given by:
V[s] = (4/3)(pi)r^3
Now all we have to do is to subtract the volume of the sphere from the volume of the cylinder, or V[c]-V[s] to find the amount of material that must be removed from the wooden cylinder to create the largest possible sphere.
V[c]-V[s] = 2(pi)r^3 - (4/3)(pi)r^3 = (2/3)(pi)r^3