“The All-Purpose Calculus Problem”

kennedy calculus problem

A “calculus problem to end all calculus problems,” by Dan Kennedy, chairman of the math department at the Baylor School, Chattanooga, Tenn., and chair of the AP Calculus Committee:

A particle starts at rest and moves with velocity kennedy integral 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.

From Math Horizons, Spring 1994.

| Science & Math

Relativity

“I am a long time in answering your letter, my dear Miss Harriet, but then you must remember that it is an equally long time since I received it — so that makes us even, & nobody to blame on either side.”

— Mark Twain, to an autograph hunter, June 14, 1876

“My room is very easy to find when you get here, and as for distance, you know — why, Oxford is as near to London as London is to Oxford. If your geography-book doesn’t tell you that, it must be a wretched affair.”

— Lewis Carroll, to Mary MacDonald, Jan. 22, 1866

| Literature

Ice Work

http://commons.wikimedia.org/wiki/File:Ice_hockey_pictogram.svg

Three hockey pucks, A, B, and C, lie in a plane. You make a move by hitting one puck so that it passes between the other two in a straight line. Is it possible to return all the pucks to their original positions with 1001 moves?

Click for Answer
| Puzzles

In a Word

http://commons.wikimedia.org/wiki/File:Lucas_Cranach_d.%C3%84._-_Alter_Mann_und_junge_Dirnen.jpg

meretriculate
v. to deceive in the manner of a prostitute

BOW-STREET — Eliza Merchant, a black-eyed girl, of that class of women known as ‘unfortunates,’ was charged by Garnet Comerford, a sailor, with robbing him of four sovereigns, several dollars and half-crowns, and his shoes. The tar stated that on Wednesday evening, about eight o’clock he left the house of his Captain, the honourable Mr. Duncan, at the west end of town, intending to pay a visit to a sister, whom he had not seen since he left England in the Seringapatem. On the way, he met as tight a looking frigate as ever he clapt his eyes on. She hoisted friendly colours; he hove to; and they agreed together to steer into port. They sailed up the Strand, when she said she would tow him to a snug berth, and he should share her hammock for the night. He consented; and when he awoke in the morning he found that she had cut and run. His rigging had been thrown all about the room, his four sovereigns and silver, and shoes were carried off.

— The Morning Chronicle, Dec. 8, 1828

| Crime · Language · Society

Fun With Refraction

http://books.google.com/books?id=UGAvAQAAMAAJ

To show that one can focus sound waves as well as light waves, Lord Rayleigh would place a ticking pocket watch beyond the earshot of a listener, then introduce a balloon filled with carbon dioxide between them. The balloon acted as a “sound lens” to concentrate the sound, and the listener could hear the watch ticking. Rayleigh would sometimes set the balloon swaying to make the effect intermittent.

Related: Pyrex and Wesson oil have the same index of refraction — so immersing Pyrex in oil makes it disappear:

| Science & Math

Lettershifts

ETCH-PENS lettershift

When the letters in ETCH are advanced uniformly through the alphabet, they produce PENS. Likewise FUSION produces LAYOUT, INKIER produces PURPLY, SLEEP produces BUNNY, and HOTEL produces OVALS. The same technique produces the phrases SAD BEING EMPTY and MY DREAM MAN:

SAD-BEING-EMPTY lettershift

MY-DREAM-MAN lettershift

This must mean something.

| Language

Curve Stitching

http://en.wikipedia.org/wiki/File:Quadratic_Beziers_in_string_art.svg
Image: Wikimedia Commons

Mary Everest Boole, the wife of logician George Boole, was an accomplished mathematician in her own right. In order to convey mathematical ideas to young people she invented “curve stitching,” the practice of constructing straight-line envelopes by stitching colored thread through a pattern of holes pricked in cardboard. In each of the examples above, two straight lines are punctuated with holes at equal intervals, defining a quadratic Bézier curve. When the holes are connected with thread as shown, their envelope traces a segment of a parabola.

“Once the fundamental idea of the method has been mastered, anyone interested can construct his own designs,” writes Martyn Cundy in Mathematical Models (1952). “Exact algebraic curves will usually need unequal spacing of the holes and therefore more calculation will be required to produce them; it is surprising, however, what a variety of beautiful figures can be executed which are based on the simple principle of equal spacing.”

The American Mathematical Society has some patterns and resources.

| Science & Math

Podcast Episode 41: The Tragic Tale of the Lady Be Good

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The American bomber Lady Be Good left North Africa for a bombing run over Italy in 1943. It wasn’t seen again until 15 years later, when explorers discovered its broken remains deep in the Libyan desert. In this episode of the Futility Closet podcast we’ll review the strange history of the lost aircraft and trace the desperate last days of its nine crewmen.

We’ll also climb some twisted family trees and puzzle over the Greek philosopher Thales’ struggles with a recalcitrant mule.

See full show notes …

| Podcast

The Hidden Psalm

The final movement on John Coltrane’s 1965 album A Love Supreme is a “musical narration” of a devotional poem that Coltrane included in the album’s liner notes — he put the handwritten poem on a music stand and “played” it as if it were music.

“Coltrane’s hushed delivery sounds deliberately speechlike,” write Ashley Kahn in his 2003 history of the album. “He hangs on to the ends of phrases, repeats them as if for emphasis. He is in fact ‘reading’ through his horn.”

The hidden psalm was marked by New York musicians for decades before Rutgers University musicologist Lewis Porter presented a formal analysis to the American Musicological Society in 1980. “You will find that he plays right to the final ‘Amen’ and then finishes,” he writes in his 1997 biography of the saxophonist. “There are no extra notes up to that point. You will have to make a few adjustments in the poem, however: Near the beginning where it reads, ‘Help us resolve our fears and weaknesses,’ he skips the next line, goes on to ‘In you all things are possible,’ then plays ‘Thank you God’ … towards the end he leaves out ‘I have seen God.'”

“I think music can make the world better and, if I’m qualified, I want to do it,” Coltrane had said. “I’d like to point out to people the divine in a musical language that transcends words. I want to speak to their souls.”

(Thanks, Jeff.)

| Art · Language