4 A biologist is researching the growth of a certain species of hamster. She proposes that the length, \(x \mathrm {~cm}\), of a hamster \(t\) days after its birth is given by
$$x = 15 - 12 \mathrm { e } ^ { - \frac { t } { 14 } }$$
- Use this model to find:
- the length of a hamster when it is born;
- the length of a hamster after 14 days, giving your answer to three significant figures.
- Show that the time for a hamster to grow to 10 cm in length is given by \(t = 14 \ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.
- Find this time to the nearest day.
- Show that
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 14 } ( 15 - x )$$
- Find the rate of growth of the hamster, in cm per day, when its length is 8 cm .
(1 mark)