Math Chat: Water Levels And Calendars

Old water level challenge

(Thanks to Jeff Bradford.)

When you throw the anchor over the side of your boat, does the water level in the pond rise or fall? When ice melts in a glass of warm water, does the water level rise or fall?

Answers

Imagine someone on the dock first removing the anchor from the boat and then throwing it in the water. When it is lifted, the boat rises and the water level falls an amount depending on the anchor's weight. When it is thrown into the water, the level rises an amount depending on the anchor's volume. Since anchors weigh more than the same volume of water, the effect of removing the anchor from the boat is greater, and there is a net fall in water level.

Since melting ice is itself water, there is no such effect, and the water level in the glass stays about the same. Since the melting ice cools the water, the water contracts and the level falls slightly. Jan Smith concludes that "the anticipated rise of the ocean level due to global warming can therefore be only due to the ice resting on the bottom."

(Winning answers: H. Arrow, E. Brahinsky, R. Bliss, R. Cohen, A. Dunne, B. Ford, C. Gahr, J. Heine, J. Smith, L. Somers, A. Stier, and D. Trent.)

The calendar

Our current calendar divides the 365-day year into 12 months, or 52 weeks plus one extra day, which makes each successive year start on the next day of the week (or two days later after Leap Year). The Islamic calendar has shorter months, corresponding more accurately to the roughly 29-1/2 day cycle of the phases of the moon, but this calendar year is shorter than the changing seasons. The less regular Hebrew and Chinese calendars add an extra month as necessary to keep the seasons roughly on schedule.

New calendar challenge

Find a better calendar system. Are seven-day weeks best? Is there any way to avoid needing a new calendar every year?

Send answers and new questions to: Math Chat, Bronfman Science Center, Williams College, Williamstown, MA 01267;

or, by e-mail to: Frank.Morgan@williams.edu

The best submissions will receive the book "Flatland."