Applications of Applications of Exponential FunctionsExponential Functions

ObjectivesObjectives

To solve real-life application problems To solve real-life application problems using the properties of exponents and using the properties of exponents and exponential functions.exponential functions.

Compound InterestCompound Interest

Compound interest arises when interest is Compound interest arises when interest is added to the added to the principalprincipal (initial investment or (initial investment or debt), so that from that moment on, the debt), so that from that moment on, the interest that has been added also itself interest that has been added also itself earns interest. earns interest.

This addition of interest to the principal is This addition of interest to the principal is called called compoundingcompounding. .

Compound InterestCompound Interest

“Compound interest is the most powerful force in the universe.”

Albert Einstein

Compound InterestCompound Interest

PP = # of dollars (present value) = # of dollars (present value)

rr = rate= rate

nn = # of interest payments/ year = # of interest payments/ year

tt = # of years= # of years

AA = # of dollars (future value) = # of dollars (future value)

1ntr

nA P

Compound InterestCompound Interest

Suppose you deposit $9,000 in a 5% interest bearing savings Suppose you deposit $9,000 in a 5% interest bearing savings account that pays interest semi-annually (compounded twice account that pays interest semi-annually (compounded twice per year). How much money would be in the account at the per year). How much money would be in the account at the end of 9 years?end of 9 years?

2(9).05

9000 12

A

$14,036.93A

1nt

rA P

n

Compound InterestCompound Interest

Now, suppose the account is compounded quarterly. How much Now, suppose the account is compounded quarterly. How much money would be in the account at the end of 9 years? What about money would be in the account at the end of 9 years? What about after 15 years? 20 years? 32 years? after 15 years? 20 years? 32 years?

$44,137.38

32

A

after years

4(15).05

9000 14

A

$18,964.63

15

A

after years

4(32).05

9000 14

A

$24,313.36

20

A

after years

4(20).05

9000 14

A

4(9).05

9000 14

A

$14,075.49

9

A

after years

1nt

rA P

n

Compound InterestCompound InterestSuppose that in 10 years you want to have $25,000 to buy a Suppose that in 10 years you want to have $25,000 to buy a new car. If you were to open up a savings account that pays new car. If you were to open up a savings account that pays 6% interest and is compounded monthly, how much should 6% interest and is compounded monthly, how much should you deposit into the account today so that in 10 years you you deposit into the account today so that in 10 years you will be able to pay for the new car?will be able to pay for the new car?

12(10).06

25,000 112

P

12025,000 1.005P

1nt

rA P

n

25,000 (1.819)P

$13,740.82P

Compound InterestCompound Interest

Suppose you want to invest $100,000 for 6 months into an Suppose you want to invest $100,000 for 6 months into an account that compounds monthly, and at the end you want to account that compounds monthly, and at the end you want to have $102,000. What does the account’s interest rate need to have $102,000. What does the account’s interest rate need to be?be?

0.0397 3.97%r or

6 1.02 112

r

12(0.5)

102,000 100,000 112

r

1nt

rA P

n

6

1.02 112

r

0.003312

r

Compound InterestCompound Interest

Now suppose instead you put it into an account that Now suppose instead you put it into an account that compounds continuously. What would the interest rate compounds continuously. What would the interest rate need to be for that $100,000 to become $102,000 in 6 need to be for that $100,000 to become $102,000 in 6 months?months?

0.0396 3.96%r or

(0.5)ln1.02 ln re

(0.5)102,000 100,000 rertA Pe(0.5)1.02 re

0.0198 (0.5)r