Pole Position, a retailer at Destiny Mall, has a variable cost of $$2$ per lap driven. It has identified

two segments of customers: Hard$-$Core drivers and Just$-$For$-$Fun drivers. For simplicity,

throughout this problem, assume there is exactly one customer in each of the two segments. Market

research has revealed how each segment values the experience, depending on how many laps are

raced:

Suppose Pole Position charges a price of $$10$ per lap.

a$)$ How many laps would a Hard$-$Core customer purchase? Hint: for each possible # of

Laps, calculate the total price paid and the net benefit, i$.$e$.,$ Total Benefit less Total Price.

The customer will buy the number of laps that yields the highest net benefit.

b$)$ How much profit would Pole Position earn from these sales?

c$)$ How many laps would a Just$-$For$-$Fun customer purchase?

d$)$ How much profit would Pole Position earn from these sales?

Suppose Pole Position only had one Hard$-$Core driver and no Just$-$For$-$Fun drivers.

a$)$ What price per lap would maximize its profit?

b$)$ At this price, how many laps would a Hard$-$Core driver purchase?