I also came across this problem from Datagenetics, which also did an analysis of actual cans. I learned about this fact from Aha! Solutions by Martin Erickson. This means the ratio of the can’s height to its radius should equal 2 to give a minimum surface area for the can. We can therefore set those two equations equal to each other to solve for h.
We know that for any can the volume is V = π r 2 h. This is the formula for the volume with minimum surface area. Notice the second derivative is positive (4π + 4 V/ r 3 > 0 for r > 0), so this is a minimum value. In order to find the minimum, we can take the derivative with respect to the radius r and set it equal to 0. The surface area can be made arbitrarily large by increasing the radius of the can, so there is no maximum value.
We can substitute this into the surface area equation to express the surface area as a function of the radius. The can’s surface area is equal to the sum of the area of two circles (the top and bottom of can) and the area of the side, which is a rectangle with one dimension h and the other dimension the circumference of the top of the can.įor a fixed volume V, we can solve for the height of the can:
#4.1 PUZZLE DIMENSION FREE#
MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.Īnswer To Which Can Size Is Best? The Optimal Cylinder Riddleįor a can with radius r and height h, the volume is equal to: "All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. What’s The Best Soup Can Size? The Optimal Cylinder Puzzle Of all the different shapes, which one has the least surface area? That is, which can uses the least material to contain the volume? Solve for the ratio of the height to the radius. The cylinder can either be wide and short (a large radius r and a small height h) or narrow and tall (a small radius r and a large height h). There are many possible shapes for the can that could hold a volume of V. Imagine you want a cylindrical can to contain a fixed volume V. This is a problem from Math Puzzles Volume 2 that I had not posted about before. (rated 4.2/5 stars on 22 reviews) Kindle Unlimited Math Puzzles Volume 3 is the third in the series. Math Puzzles Volume 2 is a sequel book with more great problems. Volume 1 is rated 4.4/5 stars on 87 reviews. Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory. Mind Your Puzzles is a collection of the three “Math Puzzles” books, volumes 1, 2, and 3. Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.2/5 stars on 76 reviews)
#4.1 PUZZLE DIMENSION HOW TO#
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