### 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.

### 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?

### 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 2

##### Stage: 5 Challenge Level:

We then solve this equation: \begin{align*} \frac {\mathrm{d}N}{\mathrm{d}t}&=rN(t) \\ \frac {\mathrm{d}N}{N(t)}&=r\mathrm{d}t \\ ln\big(N(t)\big)&=rt+c \\ N(t)&=e^{rt}e^{c} \\ \therefore N(t)&=N_0e^{rt} \end{align*}

• The birth and death rates remain constant and unvaried for different individuals
• No random changes over time (eg. due to fire, drought)
• The population is closed
• No time lag in the continuous model