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Exponential Growth and Decay

Exponential Growth

Any exponential growth can be simulated by the general formula ( )

Where ( ) is the growth at time t

is the initial amount

is the growth constant

Example

A bacteria colony is experiencing rapid and uninhibited growth. There are initially 20 cells of

bacteria in the colony. After 1 hour the colony is made up of 120 cells. How large will the

colony be after 12 hours?

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Exponential Decay

Any exponential decay can be simulated by the general formula ( )

Where ( ) is the decay at time t

is the initial amount

is the decay constant

Example

One of the by-product of nuclear power generation is Uranium-233. Uranium has a half-life

159200 years. If 120 pounds are produced from a power plant, how long before it degrades to

less than one pound?

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40

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Practice

1. There were 20 rabbits on an island. After six months the number of rabbits had increased

to 100. If the number of rabbits increased exponentially, then how many rabbits will there

be at the end of one year?

2. In mediaeval times there were 10,000 people living in a city that was struck by a plague

so that people began to die at an exponential rate daily. After 6 days, there were only

8,500 people living. How many were living after three weeks?

3. A scientist started with a culture of 20 bacteria in a dish. The number of bacteria at the

end of each successive hour increased exponentially, so that the number at the end of one

day was 220. To the nearest million, how many bacteria were there after one week?

4. The number of people living in a country is increasing each year exponentially so that the

number of people 5 years ago was 4 million. The number of people in five years time is

projected to be 6.25 million. What is the present population of the country?

5. Jamal bought a new car for $30,000. After two years the value of the car had depreciated

so that it was then only worth $20,000. If the value of the car each year decreases

exponentially, then (to the nearest dollar) how much will it be worth after another 5

years?

6. There were 80 dodos living on an island. After five years the number of dodos had

decreased to 55. The number of dodos alive each year decreased exponentially. How

many more years (to the nearest year) was it before there were just two dodos left on the

island?

7. 6 years ago Rosa had $2,000 in the bank. She now has $4,500. The amount of money at

the end of each year increases exponentially. How many more years (to the nearest year)

will pass before Rosa has $20,000 in the bank? (Assume that she doesn't deposit or

withdraw any money.)

8. A biologist is researching a newly-discovered species of bacteria. At time t = 0 hours, he

puts one hundred bacteria into what he has determined to be a favorable growth medium.

Six hours later, he measures 450 bacteria. Assuming exponential growth, what is the

growth constant "k" for the bacteria? (Round k to two decimal places.)

9. A certain type of bacteria, given a favourable growth medium, doubles in population

every6.5 hours. Given that there were approximately 100 bacteria to start with, how many

bacteria will there be in a day and a half?

10. Radio-isotopes of different elements have different half-lives. Magnesium-27has a half-

life of 9.45 minutes. What is the decay constant for Magnesium-27? Round to five

decimal places.