6 The length of time, \(T\) minutes, that a passenger has to wait for a bus at a certain bus stop is modelled by the probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { 4000 } \left( 20 t - t ^ { 2 } \right) & 0 \leqslant t \leqslant 20
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(y = \mathrm { f } ( t )\).
2. Hence explain, without calculation, why \(\mathrm { E } ( T ) = 10\).
3. Find \(\operatorname { Var } ( T )\).
4. It is given that \(\mathrm { P } ( T < 10 + a ) = p\), where \(0 < a < 10\). Find \(\mathrm { P } ( 10 - a < T < 10 + a )\) in terms of \(p\).
5. Find \(\mathrm { P } ( 8 < T < 12 )\).
6. Give one reason why this model may be unrealistic.
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