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Exponential Equations: Exponential Growth and Decay Application

A common application of exponential equations is to model exponential growth and decay such as in populations, radioactivity and drug concentration.

The formula for exponential growth and decay is:

EXPONENTIAL GROWTH AND DECAY FORMULA

y = abx

Where a ≠ 0, the base b ≠ 1 and x is any real number

In this function, a represents the starting value such as the starting population or the starting dosage level.

The variable b represents the growth or decay factor. If b > 1 the function represents exponential growth. If 0 < b < 1 the function represents exponential decay.

When given a percentage of growth or decay, determined the growth/decay factor by adding or subtracting the percent, as a decimal, from 1.

In general if r represents the growth or decay factor as a decimal then:

b = 1 – r     Decay Factor
b = 1 + r     Growth Factor

A decay of 20% is a decay factor of 1 – 0.20 = 0. 80

A growth of 13% is a growth factor of 1 + 0.13 = 1.13

The variable x represents the number of times the growth/decay factor is multiplied.

Let’s solve a few exponential growth and decay problems.

POPULATION

The population of Gilbert Corners at the beginning of 2001 was 12,546. If the population grew 15% each year, what was the population at the beginning of 2015?

Step 1: Identify the known variables.

Remember that the decay/growth rate must be in decimal form.

Since the population is said to be growing, the growth factor is b = 1 + r.

y = ? Population 2015

a = 12,546    Starting value

r = 0.15 Decimal form

b = 1 + 0.15 Growth Factor

x = 2015 – 2001 = 14    Years

Step 2: Substitute the known values.

y = abx

y = 12,546(1.15)14

Step 3: Solve for y.

y = 88,772

RADIOACTIVITY

Example 1: The half-life of radioactive carbon 14 is 5730 years. How much of a 16 gram sample will remaining after 500 years?

Step 1: Identify the known variables.

Remember that the decay/growth rate must be in decimal form.

A half-life, the amount of time it takes to deplete half the original amount, infers decay. In this case b will be a decay factor. The decay factor is b = 1 – r.

In this situation x is the number of half-lives. If one half-life is 5730 years then the number of half-lives after 500 years is x=5005730x=5005730

y = ? Remaining grams

a = 16 Starting value

r = 50% = 0.5    Decimal form

b = 1 – 0.5 Decay Factor

x=5005730x=5005730 No. of Half lives

Step 2: Substitute the known values.

y = abx

y=16(0.5)5005730y=16(0.5)5005730

Step 3: Solve for y.

y = 15.1 grams

DRUG CONCENTRATION

Example 2: A patient is given a 300 mg dose of medicine that degrades by 25% every hour. What is the remaining drug concentration after a day?

Step 1: Identify the known variables.

Remember that the decay/growth rate must be in decimal form.

A drug degrading infers decay. In this case bwill be a decay factor. The decay factor is b = 1 – r.

In this situation xis the number of hours, since the drug degrades at 25% per hour. There are 24 hours in a day.

y = ? Remaining drug

a = 300 Starting value

r = 0.25 Decimal form

b = 1 – 0.25   Decay Factor

x = 24 Time

Step 2: Substitute the known values.

y = abx

y = 300(0.75)24

Step 3: Solve for y.

0 = 0.30 mg