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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.

Air Routes

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

Over-booking

The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?

Quorum-sensing

Age 16 to 18 Short Challenge Level:

The bacteria divide twice an hour, so divide 48 times in 24 hours.
 
Thus $2^{48}$ bacteria cells after 24 hours, assuming the bacteria split in two when they divide.
 
X has half life of 10 mins, so there are $6\times 24$ half lives in 24 hours. We only consider the bacteria released near the end then, as very little from the beginning will be left after 24 hours. 
 
Now $10^{11}< 2^{48}$ so if Rudolph's nose was 4mL, his nose will certainly be glowing.
 
Assuming instead that Rudolph's nose was 4mL, we need the concentration of bacteria to be $4\times 10^{11}$.

After 23.5 hours there were $2^{47}$ cells and $2^{47}$ have just been released. Right at 24 hours, there will be $\frac {2^{47}}{2}+2^{48}$ cells, which is greater than $4\times 10^{11}$.
 
So Rudolph will have a glowing nose.