N2010 P1 Q7

A bottle containing liquid is taken from a refrigerator and placed in a room where the temperature is a constant \(20{}^\circ \mathrm{C}\). As the liquid warms up, the rate of increase of its temperature \(\theta {}^\circ \mathrm{C}\) after time \(t\) minutes is proportional to the temperature difference \(\left( 20-\theta \right){}^\circ \mathrm{C}\). Initially the temperature of the liquid is \(10{}^\circ \mathrm{C}\) and the rate of increase of the temperature is \(1{}^\circ \mathrm{C}\) per minute. By setting up and solving a differential equation, show that \(\theta =20-10{{e}^{-\frac{1}{10}t}}\)[7]

Find the time it takes the liquid to reach a temperature of \(15{}^\circ \mathrm{C}\), and state what happens to \(\theta \) for large values of \(t\). Sketch a graph of \(\theta \) against \(t\).[4]