## Estimating

Problems and tasks from a variety of sources intended to illustrate the way mathematics arises in life and work.

Rules of Thumb. Here are two rules of thumb for converting Centigrade to Fahrenheit: "Double C and add 30;" or "Double C, subtract 10%, then add 32." How well do these work? Which is more suitable to the northern states, which to southern?

Tailgating. Compare via calculation, formula, and graphs two common rules for safe driving: to stay one car length behind the car ahead for every 10 mph, or stay two seconds behind the car ahead.

Digitizing Libraries. Estimate how many words there are in all the books in the school library. Then figure out how many megabytes of disk storage would be required to store the entire library on a computer. Finally, if all these books were digitized, how long would it take to transmit the entire contents of the library via a 56KB computer network?

Bumps. Suppose an orange were blown up to the size of the earth? How high would be the tallest mountains? Or, imagine the earth shrink the size of an orange. Which would be smoother?

Estimating Area. Estimate the areas of several states of different shapes, given only a map (with scale) of each state. Compare with accurate values (from an almanac) to see how the the shapes of the states affest the accuracy of the estimates.

Fermi Estimates. "Back-of-the-envelope" estimates are based on rough approximations that can be derived from common sense or everyday observation. Examples:

• How many school teachers are there in New York City? How many electricians? How many morticians?
• How many gallons of water are there in Lake Superior? How does that compare with the amount of water used by U. S. households each year?
• How many miles of streets are in your city or town?
• What is your share of the national debt?
• When will your town fill up its landfill?
• How far do you walk if you dance all night?

Order of Magnitude. Estimate how many cups of water there are in Lake Superior.

Making Estimates. Each year visitors to Disney World eat 7 million hamburgers, 5 million hot dogs, 5 million pounds of French Fries, and 46 million Coca Cola drinks. What can you infer from these data about the likely number of visitors?

Mental Arithmetic. Using only mental arithmetic (no fingers, pencil or a calculator), double and halve any integer or simple fraction.

Restaurant Bill. Without pencil or calculator, mentally round off a restaurant bill in order to estimate to the nearest 50 cents the amount of a 15% tip.

Estimating Volume. A large load of topsoil forms a conical pile. Because of its size, you cannot directly measure either its diameter or its height. Find a strategy for estimating how many truck-loads of topsoil it contains.