### 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?

# Spinners

##### Stage: 5 Challenge Level:
Use the interactivity to experiment. Start with $2$ spinners labelled $1, 2, 3$ and $4$. Then each spinner is represented by the polynomial $x + x^2 + x^3 + x^4$. The computer will model your experiment by randomly choosing one number from each spinner and recording the sum of the two numbers. Repeat the experiment enough times for the freqency distribution of the scores to be close to the theoretical distribution.

Look at the coefficients when you have expanded the polynomial $(x + x^2 + x^3 + x^4)^2$ and compare them to the relative frequencies in which the different possible scores occur.

Now combine the factors of the polynomial in different ways. In each polynomial factor the powers correspond to the labels on the spinners so that different factorisations correspond to different labelings of the spinners.

Now do a computer experiment with spinners with the new labelling.

Because the expanded polynomial is the same the differently labelled spinners will produce scores with the SAME relative frequencies. Try it out.