2022 SAJC P1 Q7

The rate of temperature loss of an animal corpse can be estimated using Newton’s Law of Cooling, which states that the rate of change of temperature \(\theta {}^\circ \mathrm{C}\), \(t\) hours after death of an animal is proportional to the difference between its body temperature \(\theta {}^\circ \mathrm{C}\) and the constant surrounding temperature \({{\theta }_{0}}{}^\circ \mathrm{C}\), where \(\theta >{{\theta }_{0}}\).

1. Write down a differential equation for this situation. Solve this differential equation and show that the general solution of the above differential equation is given by \(\theta ={{\theta }_{0}}+A{{e}^{-kt}}\), where \(A\) and \(k\) are positive constants.[3]

It is given that \({{\theta }_{0}}=24\), the initial value of \(\theta \) is \(36\) and the initial rate of temperature loss is \(2.5{}^\circ \mathrm{C}\) per hour.

2. Calculate the exact values of \(A\) and \(k\).[3]
3. Hence, sketch the graph of \(\theta \) against the \(t\).[2]
4. Explain why the rate of change of temperature of an animal corpse cannot be modelled by a constant rate of decrease of \(1.5{}^\circ \mathrm{C}\) per hour.[1]