10 Liquid is flowing into a small tank which has a leak. Initially the tank is empty and, \(t\) minutes later, the volume of liquid in the tank is \(V \mathrm {~cm} ^ { 3 }\). The liquid is flowing into the tank at a constant rate of \(80 \mathrm {~cm} ^ { 3 }\) per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to \(k V \mathrm {~cm} ^ { 3 }\) per minute where \(k\) is a positive constant.

1. Write down a differential equation describing this situation and solve it to show that $$V = \frac { 1 } { k } \left( 80 - 80 \mathrm { e } ^ { - k t } \right)$$
2. It is observed that \(V = 500\) when \(t = 15\), so that \(k\) satisfies the equation $$k = \frac { 4 - 4 e ^ { - 15 k } } { 25 }$$ Use an iterative formula, based on this equation, to find the value of \(k\) correct to 2 significant figures. Use an initial value of \(k = 0.1\) and show the result of each iteration to 4 significant figures.
3. Determine how much liquid there is in the tank 20 minutes after the liquid started flowing, and state what happens to the volume of liquid in the tank after a long time.