Warm Up

If you invested $10,000 in a savings account that pays 5.5% interest compounded quarterly how much money would you have after 6 years?

HW Check 5.3

Exponential Growth&

Exponential Decay

Many real world phenomena can be modeled by functions that describe how things grow or decay as time passes. Examples of such phenomena include the studies of populations, bacteria, radioactive substances, electricity, temperatures and credit payments.

Any quantity that grows or decays by a fixed percent at regular intervals is said to possess exponential growth or exponential decay.

Exponential Growth vs. Decay

Exponential Growth

b > 1

Exponential Decay

b < 1

6

4

2

-2

-5

f x = 2x

8

6

4

2

-5 5

h x = 0.5x

y = a∙bx a – starting amount b – growth or decay factor

Growth or Decay…Why?

y = 5 ·2 x

y = 4 ·0.87 x

y = 10 ·0.02 x

y = 6 ·1.5 x

1) A population of 300 ants increases by a growth factor of 2 every month. How many ants will there be in 3 months?

2) The value of a $3000 diamond increases by a growth factor of 1.2 every 5 years. How much is the diamond worth in 10 years?

Exponential Growth & Decay Models

Growth: Y = a(1 + r)x

Decay: Y = a(1 – r)x

a = initial amountr = growth/decay rate (often a

percent)x = number of time intervals that

have passedY= final amount

(1 + r) is called the growth factor

(1 – r) is called the decay factor

Example #3

The value of an iPad decreases at 35% per year. If the starting price of the iPad is $500, write the exponential function.

y = 500 · 0.65 x

Make a table of values to find the value of your iPad for the domain {0,1,2,3,4,5}, how much will your iPad be worth after 5 years?

# years iPad0 500

1 325

2 211.25

3 137.31

4 89.25

5 58.02

Example #4

Suppose the acreage of forest is decreasing by 2% per year because of development. If there are currently 4,500,000 acres of forest, determine the amount of forest land inv5 years.

y = a · bx

Starting Value (a)a = 4,500,000

Exponential Decay Functiony = a · bx

y = 4,500,000 ·0.98 xAfter 5 Years

y = 4,500,000 ·0.98 5

Half-Life Problems

Some unstable substances such as plutonium decay over time.

To measure the rate at which they decay scientists refer to their “half life.”

The half-life is the time it takes for half the initial amount of the substance to decay

4) The pesticide DDT was widely used in the United States until its ban in 1972. DDT is toxic to a wide range of animals and aquatic life, and is suspected to cause cancer in humans.

Write an equation to examine the 15 year half-life of 100 grams of DDT.

How much DDT would be remaining in 45 years?