Exponential Decay/Growth - Resources

The following information will help you find exponential growth and decay.

  • exponential decay/growth formula:
    a = initial (beginning amount)
    b = growthTo find the growth factor, change percent to a decimal and add to 1. or decayTo find the decay factor, change percent to a decimal and subtract from 1 if substance has a half-life, that is exponential decay. factor
    x = time
    y = amount of substance after a given period of time

 

Example 1: Exponential Decay

Phosphorus-32 is used to study a plant's use of fertilizer. It has a half-life of 14.3 days. Write an exponential decay function for a 70-mg sample. How much phosphorus-32 remains after 72 days (round answer to the nearest tenth).

y = abx

Exponential function: y = 70( )^(x/ )

Answer: y = 70(0.5)^(x/14.3)

 

Amount after 72 days: y = 70(0.5)^( /14.3 )

Answer: y = 70(0.5)^(72/14.3)

 

y = mg

Answer: y = 2.1 mg

 

Example 2: Exponential Decay

Carbon-14 is used to determine the age of artifacts in carbon dating. Carbon-14 has a half-life of 5730 years. Write an exponential decay function for a 52-mg sample. Find the amount of carbon-14 remaining after 20,000 years. (Round answer to nearest tenth.)

y = abx

Exponential function: y = 52( )^(x/ )

Answer: y = 52(0.5)^(x/5730)

 

Amount after 20,000 years: y = 52(0.5)^( /5730 )

Answer: y = 52(0.5)^(20000/5730)

 

y = mg

Answer: y = 4.6 mg

 

Example 3: Exponential Decay

Country A has been experiencing a decline in population. In 1980, the population was 250,000 and it is decreasing at 4% per year. Approximately what will the population be in 1990? Round answer to the nearest person.

y = abx

a =

b =

x = number of years since 1980: 1990 - 1980 =

Answer:

a = 250,000

b = 0.96

x = 10

 

Now, solve. Round your answer to the nearest cent.

y = abx = 250,000(0.96)10 =

Answer = 166,208

 

 

Example 4: Exponential Growth

The population of a popular town in 2003 was approximately 35,000 people with an annual rate of increase of about 2.4%.

a) What is the growth factor for this town?

Answer: 1.024. You can find the growth factor by converting the annual rate of increase to a decimal and adding it to 1.

 

(b) Write an equation that would model future growth for this town.
y = abx

y = ( )x

Answer: y = 35000(1.024)x

 

(c) Estimate the population in 2008 to the nearest hundred people.

y = 35,000(1.024)5 ≈

Answer: 39,400

 

 

Example 5: Exponential Growth

A certain strain of bacteria doubles every 4 minutes. Assuming that you start with only one bacterium, how many bacteria could be present at the end of 84 minutes?

a) The initial amount of bacteria is 1.
b) Since the bacteria is doubling, the growth factor is 2.
c) Write an equation to model the bacteria growth over a given period of time.

y = ( )^(x/ )

Answer: y = 1(2)^(x/4)

 

d) To the nearest bacterium, how many bacteria are present after 84 minutes?

y = 1(2)^(84/4) = bacteria

Answer: 2,097,152 bacteria

 

 

 

 

Next Page