# Frank Morgan's Math Chat - "AVERAGE" TEMPERATURE

April 20, 2000

OLD CHALLENGE. The National Weather Service approximates the daily average temperature by the average of the daily high and low temperatures. Why is this such a good approximation?

ANSWER. It is not a good approximation on a warm day when the temperature suddenly plummets at 11:59 pm. Since such drastic and precisely timed events are rather rare, it is usually a pretty good approximation.

Mathematically, the approximation is good because Earth's warming and cooling behave similarly, as pointed out by John M. Sullivan. The approximation would be worse if the Earth warmed up linearly and cooled off exponentially, like a pot of tea.

QUESTIONABLE MATHEMATICS. Eric Brahinsky reports that "the "Ripley's Believe It or Not!" feature, published in the San Antonio Express-News on 04/10/00, claimed:

Believe It or Not! TWO OFFICE WORKERS SURVIVED AFTER THE ELEVATOR THEY WERE RIDING INSIDE New York City's EMPIRE STATE BUILDING FELL 40 STORIES AT A RATE of 1,395.5 ft. PER SECOND - BEFORE FINALLY STOPPING ON THE 4TH FLOOR!"

Brahinski estimates that 40 stories is about 500 feet. Even if the elevator were in free fall the whole way, its final velocity would be just the square root of twice the acceleration (32 ft/sec/sec) times the distance, or only about 180 feet per second. Brahinsky concludes, "So unless some information was omitted (e.g., pranksters had affixed a rocket engine to the top of the elevator...), I think Mr. Ripley had better do his homework."

For another question on falling elevators, see Math Chat of February 18, 1998.

Readers are invited to send in more examples of questionable mathematics.

NEW CHALLENGE (Joe Shipman). On ABC TV's "Who Wants to be a Millionaire," after winning $250,000, you are guaranteed to keep$32,000, and you go on to $500,000 and$1,000,000 questions. After hearing a question, you can answer correctly and win the new amount, walk away with your previous winnings, or answer incorrectly and leave with $32,000. How sure should you be of your answer to the$500,000 question to answer (to maximize your expected winnings). Assume no "lifelines" (opportunities for special outside help) remain.

Send answers, comments, and new questions by email to Frank.Morgan@williams.edu, to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.

THE MATH CHAT BOOK, including a \$1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).