The temperature \(\theta\) °C of a room \(t\) hours after a heating system has been turned on is given by
$$\theta = t + 26 - 20e^{-0.5t}, \quad t \geq 0.$$
The heating system switches off when \(\theta = 20\). The time \(t = \alpha\), when the heating system switches off, is the solution of the equation \(\theta - 20 = 0\), where \(\alpha\) lies in the interval \([1.8, 2]\).
- Using the end points of the interval \([1.8, 2]\), find, by linear interpolation, an approximation to \(\alpha\). Give your answer to 2 decimal places. [4]
- Use your answer to part (a) to estimate, giving your answer to the nearest minute, the time for which the heating system was on. [1]