7 A quantity of oil is dropped into water. When the oil hits the water it spreads out as a circle. The radius of the circle is \(r \mathrm {~cm}\) after \(t\) seconds and when \(t = 3\) the radius of the circle is increasing at the rate of 0.5 centimetres per second.
One observer believes that the radius increases at a rate which is proportional to \(\frac { 1 } { ( t + 1 ) }\).

1. Write down a differential equation for this situation, using \(k\) as a constant of proportionality.
2. Show that \(k = 2\).
3. Calculate the radius of the circle after 10 seconds according to this model. Another observer believes that the rate of increase of the radius of the circle is proportional to \(\frac { 1 } { ( t + 1 ) ( t + 2 ) }\).
4. Write down a new differential equation for this new situation. Using the same initial conditions as before, find the value of the new constant of proportionality.
5. Hence solve the differential equation.
6. Calculate the radius of the circle after 10 seconds according to this model.