Mathematics 9709 · AS & A Level · Continuous random variables
Continuous random variables — practice question
Bottles of Lanta each hold about $300$ ml of juice. If the juice volume, in millilitres, in a bottle is $300 + X$, then $X$ is a random variable with probability density function defined by
$f(x) = \begin{cases} \frac{3}{4000}(100 - x^2), & -10 \leq x \leq 10, \\ 0, & \text{otherwise}. \end{cases}$
(a)[3]
Determine the probability that a randomly selected bottle of Lanta contains more than $305$ ml of juice.
(b)[4]
Given that $25\%$ of bottles of Lanta contain more than $(300 + p)$ ml of juice, show that
$p^3 - 300p + 1000 = 0$.
(c)[2]
Given that $p = 3.47$, and that $50\%$ of bottles of Lanta contain between $(300 - q)$ and $(300 + q)$ ml of juice, determine $q$. Give a reason for your answer.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Try to evaluate $\int (100 - x^2)\,dx$.” …