2024 SAJC P1 Q10

Scientists are investigating the growth in length of a particular species of fish.

1. The scientists used the von Bertalanffy growth model to predict the length \(L\) cm of the fish at a particular time, \(t\) years after birth. It is said that the rate of growth in length of the fish is directly proportional to the difference between its theoretical maximum length of the fish, \(60\) cm and its length at the time \(t\) years. The constant of proportionality, \(k\), is known as the growth coefficient which is always positive. At birth, it is known that the rate of growth when the fish is \(10\) cm long is \(5\) cm per year.
   1. Write down a differential equation for this situation. Solve this differential equation to get \(L\) as an exact function of \(t\).[7]
   2. Find the length of the fish \(5\) years after it is born.[1]

2. Based on the data collected, one of the scientists proposed a second model to predict the length \(L\) cm of the fish at a particular time, \(t\) years after birth. The proposed model is as follows:
   \[\frac{\mathrm{d}L}{\mathrm{d}t} = \frac{50}{L}\]
   1. Given that the length of the fish is \(10\) cm long at birth, find \(L\) in terms of \(t\).[4]

   2. Comment on the validity of the second model in modelling the length of fish, \(L\) cm, at a particular time, \(t\) years after birth as compared to the model in (i).

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