2017 TJC P2 Q9

A flange beam is a steel beam with a “H”-shaped cross section, and is used as a supporting structure in construction and civil engineering. A factory manufactures both Grade \(X\) and Grade \(Y\) flange beams. The load that can be supported by a Grade \(X\) flange beam follows a normal distribution with mean \(2.43\times {{10}^{5}}\) kN and standard deviation \(4.5\times {{10}^{4}}\) kN. The load that can be supported by a Grade \(Y\) flange beam is \(1.5\) times of the load that can be supported by a Grade \(X\) flange beam.

1. Find the probability that the combined load that can be supported by two randomly chosen Grade \(Y\) flange beams is within \(1\times {{10}^{4}}\) kN of the combined load that can be supported by three randomly chosen \(X\) flange beams.[4]
2. A construction company wants to buy \(100\) sets of three Grade \(X\) flange beams. Find the probability that fewer than \(95\) of these sets can support more than \(6\times {{10}^{5}}\) kN.[3]

The company decides to place an order with the factory for a custom-made flange beam such that the probability of being able to support a load of at least \(6\times {{10}^{5}}\) kN must be at least \(0.999\). It is also assumed that the load that can be supported by the custom-made flange beam also follows a normal distribution.

3. By taking the standard deviation of a custom-made flange beam to be \(3\times {{10}^{4}}\) kN, find the smallest possible mean load in kN, giving your answer correct to the nearest thousand, for the factory to meet the company’s requirements for the custom-made flange beam.[5]