2024 CJC H1 Q12

As the weather gets hotter due to climate change, Tom sees a growing market for cold drinks. He plans to buy containers of syrup from Wholesaler A, Wholesaler B and Wholesaler C to manufacture his cold drinks for sale to customers. The volume, in litres, of syrup in the containers from the three wholesalers have independent normal distributions with means and standard deviations as shown.

1. Let \(A\) denote the volume of syrup, in litres, in one randomly chosen container from Wholesaler A. Given that \(\mathrm{P}(A > 115) = 0.15\) and \(\mathrm{P}(A > 105) = 0.85\), show that \(\mu = 110\) and find the value of \(\sigma\).[3]
2. Find the probability that the total volume of syrup in two randomly chosen containers from Wholesaler B is within \(\pm 50\) litres of the total volume of syrup in three randomly chosen containers from Wholesaler C.[4]

Syrup from Wholesaler B costs \(16\) per litre and syrup from Wholesaler C costs \(15\) per litre.

3. Find the probability that the total cost of two containers of syrup from Wholesaler B is at least \(3000\) more than three times the cost of one container of syrup from Wholesaler C.[4]

In order to increase sales to Tom, Wholesaler C decides to add a fixed \(50\) litres to each of its container of syrup and branded it as "Special".

4. State the mean and standard deviation of the volume of syrup in this "Special" container.[2]