11-5 Growth and Decay

More Mathematical Modeling

Objectives

1. solve problems involving exponential growth.2. solve problems involving exponential decay.

Exponential Growth• Exponential growth occurs when some quantity

regularly increases by a fixed percentage. • The equation for an exponential relationship is

given by

• y = C(1 + r)t

• where y represents the final amount, C represents the initial value, r represents the rate of change expressed as a decimal, and t represents time.

• An example of the equation of the last relationship above is simply y = $100 (1.05)x.

Example:In 2000, the U.S. population was 282 million.The U.S. population has been growing by about 0.8% each year.In this case, population A is growing by r % each year. After one year, population A will become

» Population + Population Increased by Rate r» A + A*r» 282 + 282*0.008

Using some algebra, we see that A + A*r = A(1 + r ).Notice that A is being multiplied by the quantity 1 + r,Numerically, 282(1 + 0.008).  

The general pattern is x y

0 A 282

1 A(1 + r) 282(1 + 0.008)

2 A(1 + r)2 282(1 + 0.008)2

3 A(1 + r)3 282(1 + 0.008)3

… … …

… … …

N A(1 + r)N 282(1 + 0.008)N

Using Excel:There are two ways to do this

year Population by adding percent

Population by multiplying by growth factor

2000 282 282

2001 =B2+B2*0.008 =C2*(1.008)

2002 =B3+B3*0.008 =C3*(1.008)

2003 =B4+B4*0.008 =C4*(1.008)

… … …

… … …

Exponential Functions

• If a quantity grows by a fixed percentage change, it grows exponentially.

• Example: Bank Account – Suppose you deposit $100 into an account that earns

5% annual interest.– Interest is paid once at the end of year.– You do not make additional deposits or withdrawals.– What is the amount in the bank account after eight

years?

Bank Account

year AmountInterest

Earned

ConstantGrowthFactor

0 $100.00 = $100.00 * 0.05 = $5.00  

1 $100.00 + $5.00 = $105.00 = $105.00 * 0.05 = $5.25 = $105.00 / $100.00 = 1.05

2 $105.00 + $5.25 = $110.25 = $110.25 * 0.05 = $5.51 = $110.25 / $105.00 = 1.05

3 $110.25 + $5.51 = $115.76 = $115.76 * 0.05 = $5.79 = $115.76 / $110.25 = 1.05

4 $115.76 + $5.79 = $121.55 = $121.55 * 0.05 = $6.08 = $121.55 / $115.76 = 1.05

5 $121.55 + $6.08 = $127.63 = $127.63 * 0.05 = $6.38 = $127.63 / $121.55 = 1.05

6 $127.63 + $6.38 = $134.01 = $134.01 * 0.05 = $6.70 = $134.01 / $127.63 = 1.05

7 $134.01 + $6.70 = $140.71 = $140.71 * 0.05 = $7.04 = $140.71 / $134.01 = 1.05

8 $140.71 + $7.04 = $147.75   = $147.75 / $140.71 = 1.05

Exponential Growth Graph

Exponential Decay

• Exponential Decay occurs whenever the size of a quantity is decreasing by the same percentage each unit of time.

• The best-known examples of exponential decay involves radioactive materials such as uranium or plutonium.

• Another example, if inflation is making prices rise by 3% per year, then the value of a $1 bill is falling, or exponentially decaying, by 3% per year.

( ) tr1×valueinitial=valuenew –

Exponential Decay: Example

• China’s one-child policy was implemented in 1978 with a goal of reducing China’s population to 700 million by 2050. China’s 2000 population is about 1.2 billion. Suppose that China’s population declines at a rate of 0.5% per year. Will this rate be sufficient to meet the original goal?

Exponential Decay: Solution

The declining rate = 0.5%/100 = 0.005

Using year 2000 as t = 0, the initial value of the population is 1.2 billion.

We want to find the population in 2050, therefore, t = 50

New value = 1.2 billion × (1 – 0.005)50

New Value = 0.93 billion ≈ 930 million

Exponential Decay

• The fixed amount of time that it takes a quantity to halve is called its half-life.

halfTt

valueinitialvaluenew/

2

1__

Example of Radioactive Decay

• Suppose that 100 pounds of plutonium (Pu) is deposited at a nuclear waste site. How much of it will still be radioactive in 100,000 years?

• Solution: the half-life of plutonium is 24,000 years. The new value is the amount of Pu remaining after t = 100,000 years, and the initial value is the original 100 pounds deposited at the waste site:

• New value = 100 lb × (½)100,000 yr/24,000 yr

• New value = 100 lb × (½)4.17 = 5.6 lb• About 5.6 pounds of the original amount will still

be radioactive in 100,000 years.

Exponential Decay Graph

Exponential Factors

• If the factor b is greater than 1, then we call the relationship exponential growth.

• If the factor b is less than 1, we call the relationship exponential decay.