Question

The city transit system carries 24800 bus riders per day for a fare of $1.85. A survey indicates that the number of riders will increase by 800 people for every $0.05 decrease in fare. What fare will produce the greatest revenue?

Answer 1

`$1.70`

A question like this implies a higher skill in algebra so not a lot of explanation given.

Explanation:

`color(blue)("Building the initial equation")`

Let revenue be `r`
Let count of change factor be `x`
Let users be `u`

`r=uxx"cost"`

Set `u=24800+800x`
Set cost `= $1.85-$0.05x`

`=>r=(24800+800x)(1.85-0.05x)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Conducting the calculations")`

Multiply out the brackets

`r=45880-1240x+1480x-40x^2`

`r=-40x^2+240x+45880`

Using completing the square

Factor out `-40`

`r=-40(x-3)^2+k+45880`

Set `-40(-3)^2+k=0 => k=+360`

`r=-40(x-3)^2+ 46240`

`x_("vertex")=(-1)xx(-3)=+3`
`r_("vertex")=46240`

The fare is `$1.85-3(0.05) = $1.70`