- In freezing temperatures, ice forms on the surface of the water in a barrel. At time \(t\) hours after the start of freezing, the thickness of the ice formed is \(x \mathrm {~mm}\). You may assume that the thickness of the ice is uniform across the surface of the water.
At 4 pm there is no ice on the surface, and freezing begins.
At 6pm, after two hours of freezing, the ice is 1.5 mm thick.
In a simple model, the rate of increase of \(x\), in mm per hour, is assumed to be constant for a period of 20 hours.
Using this simple model,
- express \(t\) in terms of \(x\),
- find the value of \(t\) when \(x = 3\)
In a second model, the rate of increase of \(x\), in mm per hour, is given by the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { \lambda } { ( 2 x + 1 ) } \text { where } \lambda \text { is a constant and } 0 \leqslant t \leqslant 20$$
Using this second model,
- solve the differential equation and express \(t\) in terms of \(x\) and \(\lambda\),
- find the exact value for \(\lambda\),
- find at what time the ice is predicted to be 3 mm thick.