Question $5-$ Quality control $(6$ marks$)$

A factory produces metal bars using a processing machine with a capacity of $100$ units per hour and the factory operates for $8$ hours per day. The record of quality control chart shows that the outputs of the machines have the average length of $\backslash (1,177\backslash$mathrm$\{$~mm$\}\backslash )$ with the standard deviation of $30$ mm $.$ The factory receives an order of the metal bars from a customer for $800$ units with the specification length of $\backslash (1,200\backslash$mathrm$\{$~mm$\}\backslash )$ with $\backslash (+/-5\backslash \%\backslash )$ tolerance. In addition, the customer has set the quality standard of the delivery batch of the products that allows a maximum of $\backslash (5\backslash \%\backslash )$ of defective products in the batch that is delivered.

Based on the above information, answer the following questions:

a$.$ Can the factory's outputs that are produced within the normal operating hours meet the quality standard set by the customers? $(2$ marks$)$

b$.$ What possible solution you can offer to the company to meet the quality standard set by the customer without changing the setting or condition of the machine? Show the calculations that support your solution and the practical implications of your solution $(\backslash (\backslash$mathbf$\{2\}\backslash )$ marks$)$

c$.$ If the factory's engineer can adjust the machine settings to change the average length of the metal bars produced $($without changing the standard deviation$),$ determine the average length that the processing machine should be set at to minimise the cost of producing the metal bars that meet the quality standard set by the customer within the normal operating hours. The cost of material is $\backslash (\backslash$$ $0\mathrm{.}10\backslash )$ per mm $.$ Show the calculations and explain your answers $(2$ marks$).$