The All-Purpose Calculus ProblemAuthor(s): DAN KENNEDYSource: Math Horizons, Vol. 1, No. 2 (Spring 1994), p. 5Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/25677962 .

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in doubt. The only question is, do his results imply Fermat's Last Theorem?

Actually, that is not the only ques tion. Given the immense amount of

publicity that has surrounded the whole

affair, what do we say if the present proof turns out to be unfixable? Or even if, contrary to everyone's expecta

tions, Fermat's Last Theorem turns out

to be false? Do we face an "image prob lem"? We probably do, but I don' t think it has anything to do with Wiles.

Many may disagree with me, but I think that, far from tarnishing our im

age as mathematicians, the events that

unfolded last summer will ultimately stand us in good stead.

There is plenty of evidence to indi cate that part of the reason why so many

people are put off by mathematics at an

To the uninitiated, it must seem as though

we have a direct line to

God.

early age is that it seems so cold, so

factual and impersonal. Mathematicians

portray themselves as a breed apart,

perfectly logical beings who always know the right trick, the appropriate substi

tution, the relevant lemma. We write

books that are heavy on fact and logic, but low on passion and personality. Just take a look at almost any mathematics book. To the uninitiated, it must seem as though we have a direct line to God.

What better way to start to dispel this

myth than for a quiet, unassuming En

glishman to think he has solved a fa mous three-hundred-year-old problem, and for the mathematical community to be so overjoyed that they talk freely to the press, write articles on the new

development, organize public lectures and distribute videotapes of the event. This is real, living mathematics for all the world to see.

DAN KENNEDY

The All-Purpose

Calculus Problem

Here's a calculus problem to end all calculus problems. (And you thought your professor assigned you hard ones!) See how many familiar themes you can find embedded in this

problem.

A particle starts at rest and moves with velocity v(t) = I e~x dx along a 10-foot

ladder, which leans against a trough with a triangular cross-section two feet wide and one foot high. Sand is flowing out of the trough at a constant rate of two cubic feet per hour, forming a conical pile in the middle of a sandbox which has been formed by cutting a square of side x from each corner of an 8" by 15" piece of

cardboard and folding up the sides. An observer watches the particle from a

lighthouse one mile off shore, peering through a window shaped like a rectangle surmounted by

a semicircle.

(a) How fast is the tip of the shadow moving? (b) Find the volume of the solid generated when the trough is rotated

about the y-axis.

(c) Justify your answer.

(d) Using the information found in parts (a), (b), and (c) sketch the curve on a

pair of coordinate axes.

^^^^

LAW*

MEutdkI / ffi/ } /[ pebbles ow (M j />^S^

DAN KENNEDY is chair of the mathematics department at the Baylor School, Chattanooga, TN and is chair of the AP Calculus Committee.

Math Horizons Spring 1994 5

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