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

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