Why We Need the FDA

C. Montgomery Burns
Springfield, USA
February 13, 2006

Math 102 Students
Wheaton College
Norton, MA 02766

Dear Calculus Students:

With all of the recent hullabaloo about nuclear materials and the ``Axis of Evil'', I've decided that the time is ripe for me to diversify my business somewhat. With the shortage of flu vaccines last fall and rising deductibles on prescriptions, the market seems perfectly situated to introduce The Itchy and Scratchy Brand over-the-counter flu remedy. When I went looking for help in determining how much to produce, your enterprising and resourceful professor naturally referred me to you.

I have a rough idea about the potential costs and revenues. There will be some small initial fixed costs of approximately \$5000 to convert a portion of my plant from handling radioactive waste to producing the flu remedy. I expect that labor costs per unit will initially decrease as we gain more efficiency, but these costs will eventually increase when we have to pay overtime. Further, we will be able to reduce the costs of materials per unit by buying in larger quantities, but we will subsequently max out the capacity of our long-time suppliers in Springfield, and costs will rise when we have to ship in materials from Capitol City.

The plan is to distribute the remedy to retailers in cases containing 1000 doses each. Upon the advice of your enterprising and resourceful professor, my top-notch R&D department (i.e. Smithers) has conducted extensive research to determine the materials costs, the labor costs, and the wholesale price the market will bear for several different levels of production.

Cases Produced 300 800 1800 2800 3800 4800 5800 6800
Materials cost per case ($) 1050 725 300 50 15 175 550 1100
Labor cost per case ($) 350 225 50 10 100 325 675 1200
Wholesale price per case ($) 10 30 60 85 100 105 105 100

There are several vital pieces of information I need in order to plan for our production. Specifically, I would like to know the number of cases we must manufacture to break even, the number to manufacture to maximize our total profit, and the number to manufacture to maximize the profit per case. I would, of course, also like to know what the profits will be in each situation. I am quite interested in your insightful analysis of this venture and your specific recommendations.

I will need your report by March 3 in order to prepare for production for next fall.

Capitalisticly yours,
C. Montgomery Burns

A Few Comments From Your Enterprising and Resourceful Professor

The information requested will almost certainly not occur at one of the given data points. I would suggest that you try to find one function that models the materials cost, a different function that models the labor cost, and another that models the wholesale price reasonably well. Plotting these points should give you an idea of the types of functions that will be likely candidates. Once you have these functions, you should be able to determine the necessary equations to find the requested information.