UNIT 5: EXPONENTIAL GROWTH AND DECAY

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CONTINUOUS Exponential Growth and Decay Percent of change is continuously occurring during the period of time

(yearly, monthly, …) Examples: “Continuous Interest”, “Radioactive Half-Life”

y = final amounta = original amountk = growth rate constantt = time+ = Growth, - = Decay

Example 1 Continuous Exponential Growth

b) Jill invested $1000 into a CD account that claims to compound continuously at 10%. How long will it take triple her investment?

a) A population of rabbits is growing continuously at 20% each month. If the colony began with 2 rabbits, how many months until they reach 10,000 rabbits?

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te 20.2000,10 te 20.5000 42.6 months

te 10.10003000 te 10.3 11.0 years

c) The population of Raleigh was 212,000 in 1990 and was 259,000 in 1998. Write a continuous exponential growth equation, where t is the numbers of years after 1990 for Raleigh’s population.

02503.0

212000259000 8

k

ek

Example 2 Continuous Exponential Decay

a) A paleontologist finds the bones of a Wooly Mammoth. She estimates the bones only contain 4% of the Carbon-14 that it would have had alive. Estimate the age of the Mammoth.

taea 00012.04. te 00012.04. 26,824.0 years old

#1: The formula for the half life of Carbon-14The half-life of Carbon-14 is 5760 years which means that every 5760

years half of the mass decays away.

taey 00012.

1b) A paleontologist finds that the Carbon-14 of a bone is 1/12 of that found in living bone tissue. What is the age of the bone?

707,2012

112

1

00012.

00012.

t

e

aea

t

t taey 00012.

#2: The half life of Sodium-22 is given below. A geologist is studying a meteorite and estimates that it contains only 12% of as much Sodium-22 as it would have when it reached the earths atmosphere. How long ago did the meteorite reach earth.

taey 2666.taea 2666.12. t2666.)12ln(.

8.0 years old

#3: Radioactive iodine decays according to the equation ,where t is in days. Find the half-life of the substance.

taey 0856.0

097.82

1 0856.0

t

e t

#4: The half life of Radium-226 is 1800 years. Find the half-life equation for Radium-226.

000385.0

5. 1800

k

e k

taey 000385.