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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.


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?

A Very Shiny Nose?

Age 16 to 18
Challenge Level

If we assume the starting time is midnight one day, and they have to set off at midnight the next day, how many bacteria will we have at 00:30?  How many at 02:00?  How many at 23:30?

Since there is a half-life of 10 minutes (which means that every 10 minutes half of the molecules of $X$ disappear), you can probably ignore most of the $X$ produced until near the end of the 24 hours.

You will need an estimate for the volume of Rudolph's nose.  You might like to model it as a sphere.  You might find this webpage helpful.