A company needs to manufacture cylindrical cans out of tin, each of which must have both a bottom and a top and a volume of 382cm3

Question

A) How should the company design the cans (i.e. what are the dimensions of the cans) if they want to minimize the cost of constructing them?

B)According to Google, most soup cans have a volume of 382cm3 with dimensions r = 3.25cm and h = 11.51cm. Does this agree with your answer to (a)? If not, give at least one reason why your answer to (a) may not be the best in the “real world”.

Answer 1

Because the cans are hollow (they must contain soup!) the amount of material used to manufacture them is given by the surface area of the cylindrical can. We want to minimize this function to make it as cheaply as possible.

Surface Area(r,h) = 2 π r h + 2 π r² , where r = radius and h = height

The surface area is a function of two variables, radius and height, but we can use our knowledge of the volume of the can to eliminate one of them.

Volume(r,h) = π r² h , where r = radius and h = height

We know that the volume must be 382 cm³, so substituting into te volume equation, and with some algebra:

h = 382 cm³ / (π r²)

We can use this equation for height to substitute into the original Surface Area function, and now have Surface Area as the function of only one variable:

Surface Area (r) = 2 π r (382 cm³ / (π r²)) + 2 π r²

or

Surface Area (r) = 764 cm³ / r + 2 π r²

Where is this function at it's lowest value? How do you find the minima of a function?