Slaq Computer Company manufactures notebook computers. The economic lifetime of a particular model is only four to six months, which means that Slaq has very little time to make adjustments in production capacity and supplier contracts over the production run. For a soon$-$to$-$be$-$introduced notebook, Slaq must negotiate a contract with a supplier of motherboards. Because supplier capacity is tight, this contract will specify the number of motherboards in advance of the start of the production run. At the time of contract negotiation, Slaq has forecasted that demand for the new notebook is normally distributed with a mean of $10,000$ units and a standard deviation of $3,500$ units. The net profit from a notebook sale is $$650($note that this already includes the cost of the motherboard, as well as all other material, production, and shipping costs$).$ Motherboards cost $$250$ and have no salvage value $($i$.$e$.,$ if they are not used for this particular model of notebook, they will have to be written off$).$

a$.(5)$ Use the news vendor model to compute the optimal order quantity of

motherboards that balances the cost of lost sales and the cost of excess material.

b$.(5)$ Now assume that the distribution follows an exponential distribution with a

mean of $0\mathrm{.}0001.$ What is the optimal order quantity now?