The T$.$ E$.$ Callarman Appliance Company is thinking of manufacturing and selling trash compactors on an experimental basis over the next $6$ months. The manufacturing costs and selling prices of the compactors are projected to vary from month to month. It cannot sell any compactors in July and it does not manufacture any compactors in December. The following table gives these forecast costs and prices.

Manufacturing Cost Selling Price $($During Month Month $$60$ $$60$ Uul August September October November December $$80$ $$60$ $$70$ $$80$ $$90$ $$60$ $$70$

All compactors manufactured during any month are shipped out in one large load at the end of that month. The firm can sell as many as $300$ units per month, but its operation is limited by the size of its warehouse, which holds a maximum of $100$ compactors. Callarman has no compactors on hand $($inventory$)$ at the beginning of July and wishes to have no compactors on hand $($inventory$)$ at the end of the test period in December.

Callarman$$s operations manager, Richard Deckro, needs to determine the number of compactors to manufacture and sell each month to maximize the firm$$s profit.

a$)$ Formulate this problem as an LP model.