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I Guess Leonardo DiCaprio Was Booked

Stan
South Park, CO 59055
March 12, 1999

Math 104 Students
Wheaton College
Norton, MA 02766
Dear Calculus Students:

Well, it's almost spring break here in Colorado -- not that you'd ever know it since it never stops snowing. I actually have some big plans for break: Cartman entered a Cheesy Poof sweepstakes, and believe it or not, the whole class won a guest shot on The Love Boat: The Next Wave. It's gonna be really cool being on the cruise ship, but I do have some issues with the script. When I went looking for help, your enterprising and resourceful professor naturally referred me to you.

There is alot about the script that I can't explain to you because I had to sign a standard non-disclosure agreement, but suffice it to say that the episode has a Titanic theme and Kenny finds himself afloat in a concrete lifeboat in the North Atlantic. According to the script, Kenny is doing fine until Cartman jumps from a flying saucer into the concrete raft and almost capsizes it.

I'm a little worried for Kenny -- after all, he hasn't had the best luck with stunts. I'm pretty sure that the boat is seaworthy (otherwise they wouldn't put Kenny in that position, would they?) but I'd like your expert analysis to ensure that it will float. Since Cartman has added a little weight lately, I'm really concerned that he may actually capsize the boat. How much would he have to weigh for the boat to go under? Also, it isn't crucial, but I would feel much more at ease if you could let me know how close the water will be to the top of the raft before Cartman jumps in.

I've included a few 2-D sketches of the boat, and after consultation with your enterprising and resourceful professor, he suggested that you may need to know that the concrete hull weighs 40 grams for each square centimeter of surface area.

I know that you are about to start your spring break, but we leave to begin filming on March 30, so I would greatly appreciate your answer by then.

Yours sincerely,
Stan


A Few Comments From Your Enterprising and Resourceful Professor

After consulting with Stan, I have a few suggestions that may help you get started:

  • You will need to use Archimedes' Principal, which states that the bouyant force on an object in water is equal to the weight of the water displaced (Calculus, Second Edition, Hughes-Hallett, Gleason, et al., 1998, pg 383).
  • The first thing to do is find the total surface area of the boat. You will need to deal with the curved ends separately from the rest of the boat. Notice that the curved ends look alot like a surface of revolution. You may need to look up the formula for the surface area of a surface of revolution in one of the Calculus books in Room 120. Ask me if you have questions about setting this up.
  • Once you have the surface area, you can find the total weight of the boat. Now you will need to find the volume of the boat to determine if the boyant force of water is enough to support the boat. If you don't know this, a cubic meter of water weighs 1000 kilograms.

Last Modified: Thursday, May 7, 2009, 7:09 AM
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