2024 VJC H1 Q13

In this question you should state the parameters of any distribution that you use.

A manufacturer produces tyres for toys. The masses in grams of one type of tyre have the distribution \(\mathrm{N}(100, 1.5^2)\).

1. Sketch the distribution for masses between \(90\) and \(110\) grams.[2]

Each tyre is attached to a randomly chosen rim. The masses in grams of the rim have the distribution \(\mathrm{N}(125, 2^2)\).

2. Find the probability that the total mass of \(4\) randomly chosen rims is more than \(502\) grams.[2]
3. Find the probability that the total mass of a randomly chosen tyre and a randomly chosen rim is between \(224.3\) and \(225.6\) grams.[3]

The cost of producing a tyre is \(\$0.02\) per gram.

The cost of producing a rim is \(\$0.03\) per gram.

4. Find the probability that the cost of producing \(10\) rims exceeds the cost of producing \(10\) tyres by more than \(\$18\).[5]