A group of private investors borrowed $$30$ million to build $300$ new luxury apartments near a large university. The money was borrowed at $6\%$ annual interest, and the loan is to be repaid in equal annual amounts over a $40-$year period. Annual operating, maintenance, and insurance expenses are estimated to be $$4,000$ per apartment. This expense will be incurred even if an apartment is vacant. The rental fee for each apartment will be $$12,000$ per year.

a$)$ Investigate the sensitivity of annual profit $($or loss$)$ to changes in the occupancy rate and provide a conclusion whether the investment should be made if the occupancy rate is expected to be $90\%.$

b$)$ Investigate the sensitivity of annual profit $($or loss$)$ to changes in the annual rental fee and provide a conclusion whether the investment should be made if the annual rental fee is expected to be $$14,000$ and the worst$-$case occupancy rate is projected to be $80\%.$