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Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

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Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

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Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

Population Dynamics - Part 4

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

So $x=0$ is stable for $-1< \lambda < 1$ and $x=1-\frac{1}{\lambda}$ for $1< \lambda < 3$

 

Oscillatory convergence to the stable fixed point.