9. What are the dimensions of a right circular cylindrical can with a given surface area which will result in the maximum volume?
This problem is interesting because food comes packed in cylindrical tin cans. Canners should be interested in how they can get the maximum amount of food into a can of given surface area
For this problem we need the formula for the volume of a cylinder
and the surface area of a cylinder
We are given that the surface area is given, that is, it is a constant. We can solve the equation in the definition of the surface area to get the height as a function of the radius.
We can now express the volume as a function of r.
To find the max's and min's, we differentiate and set the derivative = 0.
The simplest way to find the height is to take the equation
and multiply both sides by 2.
and substitute this into the definition of the surface area
which is to say that the height is equal to the diameter of the circular base.
This cylinder is also interesting because, while any cylinder can be inscribed in a sphere, only this one can have a sphere inscribed inside it. It is said that Archimedes, who discovered the formula for the volume of a sphere
is said to have had this figure inscribed on his tombstone. Since the volume of this cylinder is
the volume of the inscribed sphere is 2/3 of the volume of the cylinder. If you formed cones by joining the centers of the circular faces of the cylinder with the circle formed by the plane which is halfway between the two faces intersecting the sides of the cylinder, it can be shown that the volume of the sphere is twice the total volume of the two cylinders.
It is well known that the volume of the cones is 1/3 of the volume of the cylinder. The volume of the sphere is halfway between the volume of the cones and the volume of the cylinder.