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?

1 Answer
Mar 14, 2017

Answer:

#$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#