I Guess Leonardo DiCaprio Was Booked
South Park, CO 59055
March 12, 1999
Math 104 Students
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
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
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.
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.