Geometry Level 3

The Triangle Cookie Company in the historic district of the Old Town has been making its perfectly round cookies for decades and selling them in their signature isosceles triangle boxes, with three rows separated by straight red packing strips, following these two packaging rules:

1. All of the cookies are exactly the same size.
2. All of the cookies in the same row are tangent to each other, to the red strips, and to the sides of the boxes, with no gaps.

An enterprising employee decides to try improving the packing efficiency--measured by the ratio of the area occupied by the cookies to the area of the box--by relaxing the first rule to "All of the cookies in the same row are exactly the same size." This means that each row could have cookies of a different diameter than the other $$2$$ rows. The cookies will be packaged in an isosceles triangle box with suitable dimensions, so that it obeys the second rule.

Let $$R$$ be the packing efficiency. Then, approximately, what's the maximum possible percentage increase in $$R$$ obtained by adjusting the sizes of the round cookies instead of keeping them all the same?

×