Petrol stations very rarely run out of petrol. This is due partly to efficient deliveries but also to precise stick control.

Each type of petrol (4 star, unleaded, diesel) is stored in an underground tank and the amount left in each tank is carefully monitored using some form of dipstick.

It is easy to measure the height, say h, left in the tank.  However, the volume will be proportional to the cross-sectional area - not the height.  Suppose the cross-section is a circle (it is in fact elliptical, but a circle is a good approximation).  We will find the relationship between area, A, and height, h, and so provide a ready reckoner to convert height to area.

For simplicity, we will take r = 1m.  For values of h from 0 to 1, we will find the angle ø and the area of oil.

Problem 1   Show that cos ø = 1 - h.

Problem 2   Show that the area of the sector OAB is given by

Problem 3  Show that the area of the triangle OAB is

(1 - h) sin ø.

Problem 4   Deduce the area of the cross-section of oil and express this as a fraction, , of the complete cross-sectional area of the tank.

Problem 5   a)    Using the equation in Problem 1, find the value of ø for each value of h in the table below.

b)   Use the formula deduced in Problem 4 to find the area fractions.

 Area fraction h ø° 0 0 0 0.1 25.84 0.019 0.2 ... ... ... ... ... ... ... ... 1.0 90 0.500

Problem 6   Plot a graph of (vertical axis) against height h (horizontal axis).

Problem 7   Use your graph to estimate the height that corresponds to an area fraction of

a) 0.05         b) 0.10.

EXTENSION    Construct a dipstick for this problem, from which you can read off the area fraction against every height value.