### The Red Pill or the Blue Pill?

C. Montgomery Burns
Springfield, USA
March 27, 2006

Math 102 Students
Wheaton College
Norton, MA 02766

Dear Calculus Students:

Based on your recent recommendations, the conversion of part of my plant from handling nuclear waste to producing the Itchy and Scratchy Brand over-the-counter flu remedy (I&SBotcfr, for short) is proceeding nicely. Now we are planning for the initial product launch at major supermarkets and pharmacies nationwide. After the extraordinary consulting job you did earlier in the year, I was quite pleased when your enterprising and resourceful professor referred me to you once again.

My cracker-jack marketing department (i.e. Smithers) has determined that a fundamental part of the successful product launch will be to supply the major retail outlets with a free standing case to prominently display the I&SBotcfr. To make the case distinctive, we have decided to make the case the same rough shape as one of the I&SBotcfr capsules. The top and bottom of the case will be hemispheres, while the middle of the case is a cylinder. The top hemisphere will be an opaque blue plastic hinged cover, and the middle cylinder and bottom hemisphere will be made out of a brilliant red plastic. This is where I need your help in determining the optimal design.

The display should hold a complete case of 1000 doses of the I&SBotcfr. My top-notch R&D department (i.e. Smithers) has determined that the middle and bottom will need a volume of 0.25 cubic meters in order for this to work. Additionally, the opaque blue plastic used for the top hemisphere costs \$0.0025 per square centimeter, and the brilliant red plastic used for the middle cylinder and bottom hemisphere costs \$0.0015 per square centimeter. I need to know the optimal measurements to minimize the total materials cost for the case, as well as the total materials cost for this optimal design.

However, the opaque blue plastics market is fluctuating wildly these days. To save me from having to bother you during your final exams, I also need to know formulas for the optimal measurements and total cost if the opaque blue plastic costs \$b per square centimeter. To help me better understand these formulas, your enterprising and resourceful professor suggested that you should also give me the optimal measurements and costs at a few representative values for b.

I understand that you have an in-class activity next week, so it will be fine if I have your report by April 14.

Capitalisticly yours,
C. Montgomery Burns