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Operating a Tour Service

Date: 5/21/96 at 16:47:58
From: Anonymous
Subject: Calculus

You operate a tour service that offers the following rates:

i)  $200 per person if 50 people (the minimum number to book the 
tour) go on the tour.

ii)  For each additional person, up to a maximum of 80 people total, 
everyone's charge is reduced by $2.  It costs you $6000 (a fixed cost) 
plus $32 per person to conduct the tour.  How many people does it take 
to maximize your profit?

Date: 6/1/96 at 15:15:6
From: Doctor Anthony
Subject: Re: Calculus

Let x = number people you take.
Cost per person will be 200-2(x-50) = 200-2x+100 = 300-2x 
Total income to you = x(300-2x) = 300x - 2x^2
Total costs to you  = 6000 + 32x
   Profit = 300x - 2x^2 - 6000 - 32x
          = 268x - 2x^2 - 6000

Differentiate to find a maximum point

  dP/dx = 268 - 4x = 0 for max point.  So x = 268/4 = 67

So you should take 67 people to maximize your profit.

-Doctor Anthony,  The Math Forum
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Associated Topics:
High School Calculus

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