calculus word problem

Question

A train service carries 20,000 people daily between Rationalia and Real City. The rail line could serve up to 31,000 people. The cost to ride the train is $π, but research shows that for each fare increase of $0.02, ridership will go down by 271 people.

1. Create a function to represent the revenue of the train service after xincreases of $0.02. 
2. Given the constraints of the problem, what is the domain of your function? 
3. What fare should be charged to maximize revenue? 
4. If each rider costs the rail service $0.10, what fare should be charged to maximize profit? 

Answer 1

Let x be a multiplier

1)

Price, p(x) = π+0.02x

Quantity, q(x) = 20000-271x

Revenue, R(x) = p(x)q(x)

2) Domain

q(x) >=0

q(x) <=31000

x=[-40.59, 73.8]

3)

R'(x) = (400-271π)-10.84x ==> x = -41.683 which is outside of the domain

So use p(-40.59) ≈ $2.33

4) Cost, C(x) = 0.10q(x)=0.1(20000-271x)

Profit, P(x) = R(x)-C(x)

P'(x) = R'(x)-C'(x) = (400-271π)-10.84x -0.10(271)

x=-39.139

p(-39.139) ≈ $2.36