Measure 4 cans of various sizes. For each can, list the brand name, content, radius(in cm),
hight(in cm), volume(cubic cm) and the height/radius ratio rounded to the nearest hundredth.
Type of can Radius (r) Height (h) Volume h/r
Hunts tomato sauce 3.75cm 10.6cm 468.29cm^3 2.83
Find the dimensions that will minimize the amount of metal needed to make a can of volume V (V is a letter that represents a constant). Disregard any metal that is wasted in the manufacturing process. Be sure to show that you have minimized the amount of metal. Determine the height/radius ratio.
a) Besides the cost of the metal, you need to include the cost of manufacturing the can. To make this problem simple, we will only consider the cost of making the seam on the cylindrical part and joining the cylindrical part of the can to the top and bottom of the cans. Suppose the cost of the metal is and the joining of the seams is . Write a cost function for the making of the can.
b) Let's examine the h/r ratio for small cans and large cans. Using the cost function from Part a, find the dimensions that will minimize the cost of making a can if the volume is 100 , 200 , 300 , and 400 . Organize these dimensions and their h/r ratios in a table. You must substitute these values into your function. Otherwise, the general solution will be extremely long.
Now do the same for can volumes of 2000 , 2500 , 3000 , and 3500 . Organize this data in a separate table.
Note: You only need to show that the critical number minimizes the cost for the can whose volume is 100 (not for all 8 volumes).
c) What conclusion can you make about the most economical shape of small cans (squat, tall, slim, etc)? What conclusion can you make about the most economical shape of big cans? Are your conclusions supported or contradicted by the actual cans that you measured in Part 1 or that you see in the supermarket?