How many people will maximize the total revenue?

Question

A travel agent charges $10 for a trip to a concert if 20 people travel in a group. But for each 1 person above the 20-person level, the charge will be reduced by $0.25. How many people will maximize the total revenue for the company?

Answer 1

the total revenue = the number of people * the price per trip.

According to your information then, the number of people can be represented by 20 + x.

The price per trip goes down for each additional person, so it can be represented by 10 - .25x

 

So the total revenue = (20 + x)(10 - .25x)    multiplied out, this becomes

TR = 200 + 5x - .25x² which is a parabola opening downward.

 

If you know calculus, take the derivative, set = to zero and solve. If not, the turning point is on the axis of symmetry.

 

So the axis of symmetry is x=-b/2a or x=-5/(-.5) = 10

 

So 10 people lead to a maximum revenue of (plug it into your total revenue function) $225.

 

Hope this helps.

Answer 2

let x=the # of people and the # of $0.25 reductions

(20+x)(10-0.25x)=0

200+10x-5x-0.25x²=0

200+5x-0.25x²=0

use -b/2a

-5/(2)(-0.25)=-5/-0.5=10

10 more people will maximize the revenue, for a total of 30 people

(20+10)(10-2.50)=30*($7.50)=$225.00

29 people will give you $224.75 as will 31 people