What are the dimensions of a 12 ounce can that minimize the amount of aluminum required for the top and sides? The solution involves a sign analysis of the first derivative of surface area with respect to radius.
Also remember that soda/beer cans can be under significant pressure. Flat tops and bottoms don't lend well to pressure vessels as they will, like a soap bubble, seek spherical shape when pressurized, as a sphere is the most efficient surface area to volume container. That's why cans get smaller at the top and have a concave bottom with thicker material. Quite a lot of engineering in that container we so take for granted.
The aluminum on the tops and bottoms of soda cans are twice as thick as the sides. If you change your equation so that the surface area =(4*Pi*R^2)+(2*R*H) to represent using twice the aluminum on the tops and bottoms, then you'll get the right answer. The radius (cube root of 355/4pi) is 3.046 cm, making the height 12.18. Much closer to the 2.7cm - 3.3cm range for the radius and 12.3cm height.
A little late to the party but consider making a cost function using the surface area (i.e. the aluminum costs some value per square cm). If every part of the can costs the same amount nothing changes (besides maybe a scalar). However, it seems reasonable that the top and bottom material would want to be thicker and say, twice as expensive. This new cost function actually does optimize to something very close to a standard soda can (and of course companies want to optimize cost!).
Thank you for this video, I am doing a project on this exact topic and my math's was not really helping to clarify what I must do. But, one question to construct your graph what program did you use?
Thanks, Mark. All the slides are done in Keynote, including diagrams and almost all the animations. (I used GeoGebra for the animation around the 3 minute mark.) Recently I have started to simply rely on desmos for graphs in the videos.
Another question, what was the formula that put into geogebra to create the graph, because when I put A(r)=710/r+2 pi r^2 i do not get the same graph that you got in the video.
@@chrisodden Sir, for the graph and I was referring to the graph that appeared when you plot the function A(r)=710/r+2 pi r^2 at 3:29 and you got the graph with the curve.
It's usable when calculating material for water heaters for example. Less surface area touching the water means less heat escaping from the walls of the heater, also lower material costs.
introducing the r/h ration would lead to the same result much easier... in regards of why manufacturer do not follow this rule, it is not only it looks "bigger", but it also the dimater an avarage hand can hold... it would take just some educative facebook campaig to explain that actually the best shape is when the 2r/h ratio is equal to 1...it could save million tons of wasted material
