What is the maximum diameter of the largest discs?

What is the maximum diameter of the largest discs?

• A. 50 cm
• B. 45 cm
• C. 42 cm
• D. 40 cm

Answer: (B)

In this kind of disc packing problem, the limiting factor for how big the largest discs can be is the tightest contact constraint: the discs must fit inside the given boundary and also touch each other where the diagram shows they should. The maximum diameter occurs when all the relevant tangencies are satisfied simultaneously. If you try to make the largest discs even a little bigger, one of these tangencies would break—either a disc would overlap another or it would poke outside the boundary—so the configuration would no longer be feasible. In this particular setup, those critical tangencies are arranged so that the largest discs are just touching their neighbors and also just touching the boundary where required. Solving the geometry that enforces all those tangencies yields a diameter of 45 cm for the largest discs. Any larger diameter would violate one of the necessary contact conditions, so 45 cm is the largest possible. If you can share the diagram or the exact distances and which discs touch which boundaries, I can walk you through the exact equations step by step.

In this kind of disc packing problem, the limiting factor for how big the largest discs can be is the tightest contact constraint: the discs must fit inside the given boundary and also touch each other where the diagram shows they should. The maximum diameter occurs when all the relevant tangencies are satisfied simultaneously. If you try to make the largest discs even a little bigger, one of these tangencies would break—either a disc would overlap another or it would poke outside the boundary—so the configuration would no longer be feasible.

In this particular setup, those critical tangencies are arranged so that the largest discs are just touching their neighbors and also just touching the boundary where required. Solving the geometry that enforces all those tangencies yields a diameter of 45 cm for the largest discs. Any larger diameter would violate one of the necessary contact conditions, so 45 cm is the largest possible.

If you can share the diagram or the exact distances and which discs touch which boundaries, I can walk you through the exact equations step by step.