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.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.