Velocity and acceleration problems

4 A model helicopter takes off from a point \(O\) at time \(t = 0\) and moves vertically so that its height, \(y \mathrm {~cm}\), above \(O\) after time \(t\) seconds is given by $$y = \frac { 1 } { 4 } t ^ { 4 } - 26 t ^ { 2 } + 96 t , \quad 0 \leqslant t \leqslant 4$$

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} t }\);
      (3 marks)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
2. Verify that \(y\) has a stationary value when \(t = 2\) and determine whether this stationary value is a maximum value or a minimum value.
   (4 marks)
3. Find the rate of change of \(y\) with respect to \(t\) when \(t = 1\).
4. Determine whether the height of the helicopter above \(O\) is increasing or decreasing at the instant when \(t = 3\).

Water is being emptied out of a sink. The depth of water, \(y\)cm, at time \(t\) seconds, may be modelled by $$y = t^2 - 14t + 49 \quad\quad 0 \leqslant t \leqslant 7.$$

1. Find the value of \(t\) when the depth of water is 25cm. [3]
2. Find the rate of decrease of the depth of water when \(t = 3\). [3]