- The height of a river above a fixed point on the riverbed was monitored over a 7-day period.
The height of the river, \(H\) metres, \(t\) days after monitoring began, was given by
$$H = \frac { \sqrt { t } } { 20 } \left( 20 + 6 t - t ^ { 2 } \right) + 17 \quad 0 \leqslant t \leqslant 7$$
Given that \(H\) has a stationary value at \(t = \alpha\)
- use calculus to show that \(\alpha\) satisfies the equation
$$5 \alpha ^ { 2 } - 18 \alpha - 20 = 0$$
- Hence find the value of \(\alpha\), giving your answer to 3 decimal places.
- Use further calculus to prove that \(H\) is a maximum at this value of \(\alpha\).