Your newest venture idea is a Hyperloop service between Washington, DC$,$

and Baltimore. The Hyperloop trains will offer customers a choice between

coach and first$-$class tickets.

For the venture to comply with federal regulations, it must sell a minimum

of $10$ first$-$class tickets and a minimum of $10$ coach tickets per trip.

Currently, the profit margin is $$5$ for each coach ticket and $$8$ for each first$-$

class ticket. Due to safety reasons, the train's total capacity is $50$ travelers

$($excluding the crew$).$

While first$-$class tickets are more profitable, first$-$class seats take up more

space relative to coach seats. The overall length of the seating area of the

train is $2400$ inches. The seat pitch for the first class is $60$ inches. The

federally mandated seat pitch for the coach class is $30$ inches.

Another consideration for deciding on the allocation of the seats is the

weight capacity of the train. The allowed total passenger payload is $10,000$

lbs$.$ It is also known that first$-$class customers are, on average, heavier than

coach customers. The typical weight of a first$-$class customer is $200$ lbs$,$

while the typical weight of a coach customer is $150$ lbs$.$

a$)$ How many tickets of each type should be sold in order to maximize

profits?

b$)$ How much would Hyperloop earn over a $10-$year horizon with $365-$day

service and $100$ trains per day, assuming full utilization and assuming

that you implement your solution in part a$)?$

c$)$ Now, suppose that you could spend capital to upgrade the train in one

of two ways:

$$ Upgrade $1$: Spend $$4$ million to increase the maximum number

of travelers from $50$ to $55.$

$$ Upgrade $2$: Spend $$8$ million to increase the seating area of the

train from $2400$ inches to $2600$ inches