MATHEMATICAL METHODS CAN OPTIMISATION CALCULUS INVESTIGATION
Canned food comes in a variety of cylindrical cans: for example, narrow tall ones and wide short ones.
For a can of a fixed certain volume, if the height (h) of the can is increased, then necessarily, the radius (r) must decrease in order to maintain the fixed volume.Hence, the amount of metal required to make a can of a fixed volume depends on the exact dimensions of the can.The ratio of a can’s height to radius can be found by calculating hr and is a measure of how tall the can is compared to how wide it is. For example if hr=3 then the can is three times taller than it is wide.
You are working as an Efficiency Officer for a can manufacturing company. You are required to use differential calculus and other appropriate mathematical techniques to determine the dimensions of various cans which will optimise production costs. For the purposes of this task, the only production cost is considered to be the metal from which the can is made. Thus, optimum production cost will be achieved when surface area is optimised.
- To begin your investigation into optimising production costs, you are to investigate three different cans of fixed volumes (eg 400mL, 500mL, and 600mL) and determine the optimum radius, height, and hr ratio for each of the three cans.Include appropriate calculations, graphs and diagrams, present your findings in a table, and make use of appendices where appropriate.
- Formulate, test, and prove a conjecture regarding the hr ratio of a can (of any volume) with optimised surface area. Remember: a conjecture must be proven algebraically, stated with the heading “Conjecture”, and presented in a text box.
As the cans are to be made from rectangles and discs cut from metal (tens of thousands at a time), consideration must be given to the waste that is generated through this process. The surrounding metal can be recycled but is wasted as far as the manufacturer is concerned. For example, if the circles are cut in a certain way (as shown below), they are effectively being cut from squares and the four small sections remaining in the corners of each smaller square are wasted. This method is called ‘square packing of circles’.
- If the circles are square packed,
- calculate the percentage of metal that is wasted
- find the radius, height, and hr ratio of a 400mL can which will optimise the production cost when the circles are square packed.
Use the following as a guide to complete your calculations for part b:
- Determine a rule for the area of the base shape in terms of the radius of the circle cut from it. (In this example the base shape is a square with side 2r).
- Determine a rule for the total area of the metal required to make one can and optimise the surface area based on this rule.
- Investigatea minimum of two other methods of ‘packing circles’ and determine the method of ‘packing’, height, radius, and hr ratio which optimises the production cost of a 400 mL can (be creative). Begin by calculating the percentage waste for each of the packing methods you are investigating, and then use the guide above to complete your calculations.
- Compare the hr ratios for the calculations in Part A, Part B(1) and Part B(2) and interpret their relative sizes in the context of this investigation.
As Efficiency Officer, present your findings/ recommendations regarding the following:
- the hr ratio which optimises any can’s surface area when waste is disregarded (Part A),
- The most efficient circle packing method when waste is taken into account, and
- The optimum height, radius, and hr ratio of a 400mL can (Part B)
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