The movie theater owner discovered that there are two segments in the population that attend the shows, older $(40+)$ and younger $(21-)$ people, whose demand functions are as follows:

Older: $7000–100$p

Younger: $3000–150$p

The capacity of the theater is $2750$ seats.

Note: Demand must be calculated as an integer value.

a$.$ Find a single ticket price for both populations that maximizes contribution. Capacity is $2750.$

b$.$ Using variable pricing $($i$.$e$.,$ each population has its own price$),$ find the prices that maximize contribution for the older and younger individuals with unconstrained capacity. The incremental cost per viewer is still $$15.$ 

c$.$ Find the prices for the older and younger individuals under the capacity constraint b$=2750$ seats per show that maximize contribution. The incremental cost per viewer is still $$15.$ 

d$.$ What is the opportunity cost? What is the marginal opportunity cost.

e$.$ The theater owner offers the cheaper price tickets to the younger individuals as a service to the community. However, the tickets for the two populations are identical once they are issued. Older people have realized that if they could get a younger person to buy them tickets, they could get into the show paying less. So$,$ they try to cannibalize the theater. Assume that there is a cannibalization of about $10$ percent. Find the prices of the two populations that maximize contribution. Don$’$t forget that capacity is constrained by $2750$ seats.