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Question

For its beef stew, Betty Moore Company use aluminum containers that have the form of right circular cylinders. Find the radius and height of a container if it has a capacity of 36in^2 and is constructed using the least amount of metal

Answer 1

First of all, lets get the units of measure of the capacity of each can being 36in^3, not 36in^2.

 

 

The capacity of each can is the volume of each cylinder containing beef stew. The total surface area of the cylinder changes in relation to its height and radius. The goal is to determine the radius and height to get the minimum surface area of the can which translates to minimum amount of material for its construction. Write an equation for the total surface area of each cylinder in which is:

 

A = 2*pi*r^2 +2*pi*r*h

 

The volume of the cylinder is:

 

V = pi*r^2*h which implies h = V/(pi*r^2) = 36/(pi*r^2)

 

Now substitute h in the equation of the can's surface area as a function of r in which is:

 

A(r) = 2*pi*r^2 +2*V/r = 2*pi*r^2 + 72/r

 

The next step is to take the derivative A(r) with respect to r and set it to equal to zero. Then you solve for r in order to get the minimum surface area of the can. Hence

 

A'(r) = 4*pi*r - 72/r^2 = 0. This gives r = (18/pi)^(1/3) ≈ 1.79 in.

 

And h = 36/(pi*1.79^2) ≈ 3.58 in. These are the dimensions in which gives each can its minimum surface area.