2023 RI P1 Q6

Alice bakes a large pie and removes it from the oven at \(1\)pm. The temperature reading of the pie using a food thermometer is \(200{}^\circ \mathrm{C}\) at that instant.

She left the pie on the kitchen countertop to cool down. The temperature of the kitchen is kept at a constant \(32{}^\circ \mathrm{C}\). The temperature of the pie \(t\) minutes after it is removed from the oven is \(\theta {}^\circ \mathrm{C}\). This temperature decreases at a rate proportional to the difference between the temperature of the pie and the temperature of the kitchen.

At \(1.15\)pm, Alice checks the temperature of the pie again and it is \(180{}^\circ \mathrm{C}\).

1. Find an exact expression for \(\theta \) in terms of \(t\), simplifying your answer.[5]
2. Sketch a graph of \(\theta \) against \(t\).[2]
3. Food safety experts caution that bacteria grow fastest in food within the temperature range of \(5{}^\circ \mathrm{C}\) to \(60{}^\circ \mathrm{C}\), known as the temperature danger zone. Find the latest time to the nearest minute to safely store the pie to avoid this danger zone.[2]