3800 people buy $2 fare tickets. When the fare price raises $0.20, 65 customers are lost. If the company wants to make $10950 what must the fare be?

First, we need to write out a relationship between price and ridership. If we start out with 3800 passengers at $2 a ticket, and there is a decreasing rate that is linearly proportional to the change in price (65 per $0.20 above original price), the equation for ridership looks like this:

#r=3800-65*(p-2)/0.2#

Where #r# is the number of riders and #p# is the price per ticket. We can simplify the statement some more by simplifying the multiplication on the Right-Hand Side (RHS):

#r=3800-65/0.2*(p-2)#

#r=3800-325*(p-2)#

#r=3800-325p+650#

#color(blue)(r=4450-325p)#

Now that we have an equation for the number of riders, we would simply multiply the number of riders by the ticket price to get our answer:

#color(blue)(r)*p=10950#

#color(blue)((4450-325p))*p=10950#

#-325p^2+4450p=10950#

Notice that we now have the makings of a quadratic! Let's rearrange and simplify:

#cancel(25)(-13p^2+178p)=cancel(25)(438)#

#-13p^2+178pcolor(red)(-438)=cancel(438color(red)(-438))#

#-13p^2+178p-438=0#

Finally, lets solve using the quadratic equation:

#(-b+-sqrt(b^2-4ac))/(2a)#

#a=-13#
#b=178#
#c=-438#

#(-178+-sqrt(178^2-4(-13)(-438)))/(2*-13)#

#(-178+-sqrt(31684-22776))/(-26)#

#(-178+-sqrt(8908))/(-26)#

#(-178+-94.38220171)/(-26)#

#p=(-178+94.38220171)/(-26)#

#color(green)(p=3.216069165)#

#p=(-178-94.38220171)/(-26)#

#color(blue)(p=10.47623853#