2019 TJC P1 Q11

The daily food calories, \(L\), taken in by a human body are partly used to fulfil the daily requirements of the body. The daily requirements is proportional to the body mass, \(M\) kg, with a constant of proportionality \(p\). The rate of change in body mass is proportional to the remaining calories.

It is given that the body mass, \(M\) kg, at time \(t\) days satisfies the differential equation

\(\frac{\mathrm{d}M}{\mathrm{d}t}=k\left( L-pM \right)\),

where \(k\) and \(L\) are constants.

John’s initial body mass is \(100\)kg. Find, in terms of \(p\), the daily food calories needed to keep his body mass constant at \(100\)kg.[1]

To lose weight, John decides to start on a diet where his daily food calorie intake is \(75\%\) of the daily calories needed to keep his body mass constant at \(100\)kg.

1. Show that \(M=75+25{{\mathrm{e}}^{-pkt}}\).[4]
2. John attained a body mass of \(90\)kg after \(50\) days on this diet. If it takes him \(n\) more days to lose at least another \(10\)kg, find the smallest integer value of \(n\).[5]
3. John’s goal with this diet plan is to achieve a body mass of \(70\)kg. With the aid of a graph, explain why he can never achieve his goal.[2]
4. By considering \(\frac{{{\mathrm{d}}^{2}}M}{\mathrm{d}{{t}^{2}}}\), comment on his rate of body mass loss as time passes.[2]