A can is made out of steel for the top and bottom, and aluminium for the sides. Steel costs 3 times as much (per square cenimeter) as aluminium. What are the dimensions of the can of minimal cost, if the can is to hold 1000 cubic centimeters of fluid?
Solve for h in terms of r, given a fixed (V) volume:
Compute total cost:
Set the derivative equal to zero and solve:
(The second root is the one we want -- the other two are complex)
Now plug in V = 1000:
Consider a rectangular box -- with square cross section -- with height x, width x and length y. (Height = depth. This is the square cross section assumption). Assume that y is larger than x. The post office will not accept the box for domestic shipment unless the girth (4x) plus the length is less than or equal to 108 inches. What are the dimensions of the box that the post office will ship, that has maximum volume?
In the following we use the girth plus length restriction is used to solve to tbe length:
The Volume is then:
We now differentiate the volume:
And simplify, if yoiu like:
Set the derivative equal to 0 and solve:
The optimun is clearly x = 18
And the optimum y (length) is: