8-1: Exponential Growth8-1: Exponential Growthday 2day 2

Objective CA 12: Students know the laws of fractional exponents, understanding exponential functions, and use these functions in problems involving exponential growth and decay.

Using Exponential Growth Using Exponential Growth ModelsModels

When a real-life quantity increases by a fixed percent each year (or other time period) the amount y of the quantity after t years can be modeled by (1 )ty a r Where a is the initial amount, and r is the percent increase expressed as decimal.

The quantity 1+r is called the growth factor

Example 3: Modeling Example 3: Modeling Exponential GrowthExponential Growth

In 1980 about 2,180,000 US workers worked at home. During the next ten years, the number of workers working at home increased by 5% per year.

Write a Model giving the number of w (in millions) of workers working at home t years after 1930

(1 )

2.18(1 0.05)

2.18(1.05)

t

t

t

y a r

w

w

Graph the model

4

2

5 10 15

w = 2.181.05t

  Use the graph to estimate the year when there were about 3.22 million workers who worked at home.

1988 3.22

Compound InterestCompound Interest

Consider the initial principal P deposited in an account that pays interest at an annual rate r (expressed as a decimal), compounded n times per year. The amount A in the account after t years can be modeled by this equation:

(1 )ntr

A Pn

Example 4: Finding the Example 4: Finding the Balance of an AccountBalance of an Account

You deposit $1500 in an account that pays 6% annually. Find the balance of the account after 1 year if the interest is compounded: annually, semiannually, quarterly

110.06a) Annually 1500(1 )

11500(1.06)

1590

A

A

A

2 1

2

0.06b) Semiannually 1500(1 )

2

1500(1.03)

1591.35

A

A

A

4 1

4

0.06c) Quarterly 1500(1 )

4

1500(1.015)

1592.05

A

A

A

8 – 1 day 28 – 1 day 2Home work page 469Home work page 469

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