9 A botanist is investigating the rate of growth of a certain species of toadstool. She observes that a particular toadstool of this type has a height of 57 millimetres at a time 12 hours after it begins to grow. She proposes the model \(h = A \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)\), where \(A\) is a constant, for the height \(h\) millimetres of the toadstool, \(t\) hours after it begins to grow.

1. Use this model to:
   1. find the height of the toadstool when \(t = 0\);
   2. show that \(A = 60\), correct to two significant figures.
2. Use the model \(h = 60 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)\) to:
   1. show that the time \(T\) hours for the toadstool to grow to a height of 48 millimetres is given by $$T = a \ln b$$ where \(a\) and \(b\) are integers;
   2. show that \(\frac { \mathrm { d } h } { \mathrm {~d} t } = 15 - \frac { h } { 4 }\);
   3. find the height of the toadstool when it is growing at a rate of 13 millimetres per hour.
      (1 mark)