2025 Beatty Sec 3 EOY Q11

A particle is heated until it reaches a temperature of \(C\) degrees Celsius. It is then left to cool. Its temperature, \(T\) degrees Celsius, when it has been cooling for time \(t\) minutes, is given by the equation \(T=22+48e^{-\dfrac{t}{5}}\).

1. Find
   1. the value of \(C\),[1]
   2. the value of \(T\) after \(16\) minutes.[1]
2. The particle must be heated again if its temperature drops to \(25^\circ\text{C}\). Calculate the time taken for this to first happen.[2]
3. Sketch the graph of \(T=22+48e^{-\dfrac{t}{5}}\) for \(t\geq0\).[2]