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?

Bird-brained

Stage: 5 Challenge Level:

A mother bird has 2kg of mass available to produce her eggs. The chance of an egg surviving to hatch (which we will call p) is greater for heavier eggs.

This chance is defined by the following function:

p = 0       if individual egg mass (which we will call s) $\le{150}$g

p = $\frac{s-150}{k+s-150}$       if s > 150g, where k is a constant

Sketch a graph of this function. Can you describe how survival chance, p, is affected by egg mass, s? What assumptions have been made in applying this model?

How does the shape of the graph change as k varies? (Try thinking about what will happen if k is very big or very small)

if k = 0.4, what is the optimal size for each egg to be? Hence, how many eggs should the bird lay to produce the greatest expected number of chicks? What can you say about this number? If k was bigger, would the optimal number of eggs increase or decrease?

Given that a bird cannot actively choose how many eggs to produce and what mass each should be, do you think that birds in nature produce the optimal number of eggs? Why do you think this?