Bottles of Lanta contain approximately $$\(300 \mathrm{ml}\)$$ of juice. The volume of juice, in millilitres, in a bottle is $$\(300+X\)$$, where $$\(X\)$$ is a random variable with probability density function given by $$\[ f(x)= \begin{cases}\frac{3}{4000}\left(100-x^{2}\right) & -10 \leqslant x \leqslant 10, \\ 0 & \text { otherwise. }\end{cases} \]$$ Given that $$\(p=3.47\)$$, and that $$\(50 \%\)$$ of bottles of Lanta contain between $$\((300-q)\)$$ and $$\((300+q) \mathrm{ml}\)$$ of juice, find $$\(q\)$$. Justify your answer. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

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