A group of scientists uses Carbon-11 as a radioactive tracer to study carbon movement in plants. After the tracer has been introduced into the plant, the amount of Carbon-11, \(C\), decreases through radioactive decay at a rate equal to 2.5% of the amount present at time \(t\) minutes.
It is given that the initial amount of Carbon-11 is 123 Megabecquerel (MBq). Write down a differential equation for the decay and solve the differential equation, giving your answer in the form \(C=\mathrm{f}(t)\), where \(\mathrm{f}(t)\) is a function to be determined.[4]
Sketch the graph of \(C\) against \(t\). Explain the significance of the asymptote.[2]
The group of scientists subsequently extended the decay model by incorporating compartmental modelling. In this improved model, the plant is divided into several compartments, such as leaves, stems and roots. After introducing Carbon-11 as a radioactive tracer, the rate of change of Carbon-11 is now determined by the rate of inflowing Carbon-11, outflowing Carbon-11 and radioactive decay.
It is given that the rates of inflowing and outflowing Carbon-11 are \(te^{-t}\) and \(7\) MBq per minute respectively. Show that the differential equation corresponding to the compartmental model can be expressed in the form \[\frac{\mathrm{d}C}{\mathrm{d}t}+\lambda C=\mathrm{g}(t),\] where \(\lambda\) is a constant and \(\mathrm{g}(t)\) is a function, both of which are to be determined.[2]
By multiplying both sides of the differential equation in part (c) by \(e^{\lambda t}\) and rewriting \(\displaystyle \left(\frac{\mathrm{d}C}{\mathrm{d}t}+\lambda C\right)e^{\lambda t}\) as \(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}(Ce^{\lambda t})\), find the general solution to the differential equation in the form \(C=\mathrm{h}(t)\), where \(\mathrm{h}(t)\) is a function to be determined.[4]