Question

A commuter railway has 800 passengers a day and charges each one 2 dollars. For each 5 cents the fare is increased, 7 fewer people will ride the train. Express the income I from the train in terms of the ticket price p (in dollars).

Answer 1

Lets break the income function into two parts: We know income relies on both the cost charged per passenger, and the number of passengers.

In mathier terms Income (I) = f(cost,passengers)

More specifically, the Income equals the product of cost and the number of passengers.
If one person buys a ticket for 5 dollars, the income is 1x5 = 5$
If two people buy a ticket for 3 dollars, the income is 2x3 = 6$.

Now lets consider the cost of a ticket. The cost is said to start at 2$ so we'll let that be our constant baseline. Similarly we are given that there are 800 passengers when the ticket is 2$. Thats our other starting point.

We are also told that every time the price changes by 5 cents, 7 fewer passengers buy tickets.
So it follows that if the price is 2+.05, 800-7 people will buy tickets.
If we increase the price twice, it is 2+.05+.05=2.10$. Then 800-7-7 = 786 people will buy tickets.

To generalize this relationship, lets call the number of times we change the price 't'

We're left with the relationship that if price is 2+0.05t, the number of passengers is 800-7t

Now we can come back to our original function for income

Income = Passengers x Cost = (800-7t)*($2+$0.05t)