Question

A commuter train carries 600 passengers each day from a town to a city. A one-way trip costs 100 per person. Market research reveals that 10 fewer people would ride the train with every 1 increase in the fare. What fare should be charged to get the largest possible revenue?

Answer 1

Here the variables are

$x,$ the number of passengers,

$p,$ the price per ticket,

and $R,$ the revenue.

We have

$$R = px$$

We obtain

$$R = p(600 -10(p-100)) .$$

$$R=-10p^2+1600p, p\ge100$$

$$a=-10<0, p_v=-\dfrac{b}{2a}=-\dfrac{1600}{2(-10)}=80$$

$$R(80)=-10(80)^2+1600(80)=64000$$

The revenue has the absolute maximum with value of 64000 at price 80.