2026 ZHSS P2 Q12

The temperature, \(T^\circ\text{C}\), inside a greenhouse over a 24-hour period is modelled by

\[T=15+6\sin\left(\frac{\pi t}{12}\right)+8\cos\left(\frac{\pi t}{12}\right)\]

where \(t\) is the number of hours after 00 00 hrs.

A plant will begin flowering when the temperature first reaches \(24^\circ\text{C}\).

1. By expressing \(6\sin\left(\dfrac{\pi t}{12}\right)+8\cos\left(\dfrac{\pi t}{12}\right)\) in the form \(R\sin\left(\dfrac{\pi t}{12}+\alpha\right)\), where \(0<\alpha<\pi\), find the time, to the nearest minute, at which the plant begins flowering.[6]
2. Find the maximum temperature inside the greenhouse and state the first time at which it occurs, to the nearest minute.[3]
3. Find the value(s) of \(t\) when the temperature inside the greenhouse is \(10^\circ\text{C}\).[3]