Thursday, January 30, 2014

An area minimization problem

See the configuration in the figure, which shows a disk of radius r < 1 concentric with a unit circle. Also lying within the unit circle are two more disks, each of them tangent to both the unit circle and the disk of radius r.


Find the minimum combined area of the three disks. (That is, minimize the shaded area in the figure above with respect to the variable r.)

Two suboptimal configurations are shown below.





A variation. In the puzzle above, for each value of you will add the concentric disk of radius r plus two same-size disks that just fit within the resulting annulus. What is the minimum possible area if, for each value of r, you have to add the concentric disk of radius r plus as many same-size disks as you can just fit without overlaps inside the resulting annulus?

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