Expert: Socrates - 10/26/2010

Question
A travel agency offers an organization an all-inclusive tour for $800 per person if 100 persons or less take the tour. However, the cost per person will be reduced $5 for each person in excess of 100.  How many people should take the tour in order for the travel agency to receive the largest revenue?  What is the largest revenue?

Answer
Let x be the number of people taking the tour.
Let R(x) be the revenue.

If x<= 100 , R(x) = 800x

If x>100 , the cost per person will be 800 - (5)(x-100) = 1300-5x
Revenue is then R(x) = (x)(1300-5x) = -5x^2 + 1300x


If x<=100 , maximum revenue will be (800)(100) = $80,000

If x>100 , we want the maximum value for R(x) = -5x^2 + 1300x

The maximum for a parabola that opens down occurs when x is the average of the zeros

0 =  -5x^2 + 1300x = x(-5x+1300)

so the zeros are x = 0 and x = 260

The average is x = (0 + 260)/2 = 130

R(130) = (-5)(130)^2 + (1300)(130) = $84,500


So the largest revenue is $84,500 when 130 people take the tour