Q$1.(10$ points$)$ Town and Country Rental Car has experienced a substantial increase in business volume because of recent fare wars between the major air carriers. Town and Country operate a single office at a major international airport, with a fleet of $60$ compact and $30$ midsize cars. Recent developments have prompted management to rethink the company$$s reservation policy. The table below contains data on the rental experience of Town and Country. Midsize$-$car customers do not choose to rent a compact when no midsize car is available, or the other way around. The discount rate is available to persons who are willing to reserve a car in advance, and there is sufficient demand for the discount rate. The daily demand $($of full rate customers$)$ appears to follow a uniform distribution. The current reservation policy is that $40$ compact cars are held for customers who are willing to pay the full rate and $25$ midsize cars are held for full rate$-$paying customers. Using revenue management, determine the optimal number of compact and midsize cars to be held for customers paying the full rate.

Car Compact Midsize

Full Rate $$30$ $$45$

Discount Rate $$20$

$$30$

Daily Demand $($on the day of rental$)$ Uniform between $[40,60]$ Uniform between $[25,35]$