$35-$Page $7$a$)$ The average time before breakdown of a machine is normally distributed and has a mean of $7$ weeks and a standard deviation of $1\mathrm{.}5$ weeks. If breakdown cost averages $$2,500$ and preventive maintenance costs $5500,$ what is the optimal preventive maintenance interval? $(4$ marks$)$b$)$The frequency of breakdown of a piece of equipment per month is shown in the table. The cost of a breakdown is $$1,000$ and the cost of preventive maintenance is $$2,000$ per month. Assume that the equipment breakdown can be avoided if preventive maintenance is performed. Should the manager use preventive maintenance, or would it be better to fix the equipment when it breaks down? $(6$ marks$)$Number of breakdowns$012345$Probability$0\mathrm{.}100\mathrm{.}150\mathrm{.}200\mathrm{.}300\mathrm{.}150\mathrm{.}10$