3.03: Exponential Decay and Growth

Exponential Decay

The "Little Boy" bomb dropped on Hiroshima, Japan during Worls War II has a half-life of 703,800,000 years. The "Fat Man" bomb dropped on Nagasaki, Japan has a half-life The half-life is the time it takes for half the atoms of an element to decay. of 24,100 years.

Even now, after over half a century later, many after effects remain: leukemia, A-bomb cataracts, and cancers of thyroid, breast, lungs, salivary glands, birth defects, including mental retardation, and fears of birth defects in their children, plus, of course, the disfiguring keloid scars.

The "Little Boy" and the "Fat Man" bomb are examples of exponential decay. They are radioactive decay.

Exponential decay is related to radioactive decay, medicine dosage, and population decline.

Why do some products have a "time-date"?

Growth of bacteria in food products causes a need to "time-date" some products (like milk) so that shoppers will buy the product and consume it before the number of bacteria grows too large and the product goes bad.

This is an example of exponential growth.

Population is another example of exponential growth. As the population of the Earth grows exponentially, a strain is put on the Earth's resources.

When will the Earth's resources "run out"?

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Example

The "Fat Man" bomb was dropped on Nagasaki, Japan on August 9, 1945. If the "Fat Man" bomb contained 14 pounds of plutonium-239.

How much of plutonium-239 is still remaining today?

• Remember that the "Fat Man" bomb had a half-life of 24,100 years.
• Remember the exponential decay formula: y = ab^x
• Since the half-life is 24,100 years, x in the decay formula will be a fraction: (2013-1945=68/The half-life of the bomb)

Simplify and round your answer to the nearest hundredth of a pound.

Check Answer

• The 0.5 represents one-half of the life of something. Threfore, one-half is 0.5.

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