A laboratory synthesises organic protein filaments whose lengths depend on specific cultivation conditions. The length LLL, in millimetres (mm), of a filament is modelled by a continuous random variable with probability density function
g(l)={3250(10l−l2)0≤l≤50otherwise g(l) = \begin{cases} \frac{3}{250}(10l - l^2) & 0 \le l \le 5 \\ 0 & \text{otherwise} \end{cases} g(l)={2503(10l−l2)00≤l≤5otherwiseUse algebraic integration to determine the mean length of a filament. State your answer in millimetres and micrometres (1 mm=1000 μm1 \text{ mm} = 1000 \text{ }\mu\text{m}1 mm=1000 μm).
Show that the probability of a randomly selected filament having a length between 1 mm and 4 mm is 81125\displaystyle \frac{81}{125}12581.
A researcher examines a batch of 200 filaments grown under these conditions.
Using a suitable approximation, calculate the probability that at least 140 of these filaments have a length between 1 mm and 4 mm.