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The diagram shows the graph of the probability density function of a variable \(X\). Given that the graph is symmetrical about the line \(x = 1\) and that \(\mathrm { P } ( 0 < X < 2 ) = 0.6\), find \(\mathrm { P } ( X > 0 )\).
A flower seller wishes to model the length of time that tulips last when placed in a jug of water. She proposes a model using the random variable \(X\) (in hundreds of hours) with probability density function given by
$$f ( x ) = \begin{cases} k \left( 2.25 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1.5 0 & \text { otherwise } \end{cases}$$
where \(k\) is a constant.
Show that \(k = \frac { 4 } { 9 }\).
Use this model to find the mean number of hours that a tulip lasts in a jug of water.
The flower seller wishes to create a similar model for daffodils. She places a large number of daffodils in jugs of water and the longest time that any daffodil lasts is found to be 290 hours.
Give a reason why \(\mathrm { f } ( x )\) would not be a suitable model for daffodils.
The flower seller considers a model for daffodils of the form
$$g ( x ) = \begin{cases} c \left( a ^ { 2 } - x ^ { 2 } \right) & 0 \leqslant x \leqslant a 0 & \text { otherwise } \end{cases}$$
where \(a\) and \(c\) are constants. State a suitable value for \(a\). (There is no need to evaluate \(c\).)