P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by
replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
How much more efficiently would you be able to pack 1cm diameter disks into the 1m square? Could you make an estimate for the efficiency of packing disks of diameter 1mm?
Whilst it might seem relatively simple, the problem of 'shape packing' is often very difficult mathematically to solve with certainty for many shapes. Intuitive visualisation often works just as well as a strict mathematical analysis, and often is the only sensible possibility with packing together complicated shapes.
You might like to consider situations in which efficient shape packing is relevant in the physical world.