A company wants to sell pet food in cylindrical cans of fixed volume ml millilitres The cans have radius r cm and height The metal used to make the sides of the cans costs cent per The metal used for the top and bottom of the cans costs twice as much as that used to make the sides. The company wants to know the correct values of and which minimises the total cost of the metal used to make each can.
Recall the following formulae for cylinders: the area of the top or bottom of each can is the area of the side of each can is and the volume of each can is
a Let c denote the the total cost in cent of the metal used to make each can. Write down a formula expressing in terms of and
b Use the given information about the volume of the can to express in terms of Then combine this with your answer to part a to express the cost of each can as a function of the radius alone.
c Determine the vertical and horizontal asymptotes if any of
d Find the critical points of
e In part below you will need to sketch the graph of the function in the plane. Find the intercept of ie the root of and explain why there is no intercept.
Sketch the graph of for both positive and negative. Note: although we can study the function for negative it does not correspond to an actual cylindrical can, as these always have positive radius I
g Determine the radius which minimises the cost of each can.
