Power Point Exponential Function Applications

Applications of Exponential Functions

COMPOUND INTEREST

1nt

rA P

n

1. Cyril is being offered with an investment that promises a 10% annual interest compounded once a year. If he plans to invest Php 10, 000, how much is his money at the end of three years?

n

33

33

33

3

3

A (1 )

A (1 )

A 10,000(1 0.10)

A 10,000(1.10)

A 10,000(1.331)

A 13,310

tP r

P r

Given:P = 10,000r = 0.10n= 3

2. Find the amount if Php 3, 000 is invested at 5% annual interest rate for 8 years

(a) Compounded semi-annually

:

3,000

2

0.05

Given

P

n

r

8(2)

8

16

8

16

8

8

8

0.053,000 1

2

3,000 1 0.025

3,000 1.025

3,000 1.4845

4,454

A

A

A

A

A

b. Compounded quarterly

:

3,000

4

0.05

Given

P

n

r

8(4)

8

32

8

32

8

8

8

0.053,000 1

4

3,000 1 0.0125

3,000 1.0125

3,000 1.4881

4,464

A

A

A

A

A

3. A savings account of 5, 000 is placed at 5% per annum. How much is in the account at the end of one year if the interest is:

a. Compounded once a year (n =1)?

1nt

rA P

n

(1)(1)0.05

5000 11

5000(1.05)

5,250

A

The amount if

compounded once a year.

(4)(1)

4

For n = 4

0.05A = 5000 1+

4

= 5000(1.0125)

= 5000(1.050945)

= 5, 254.73

The amount if compounded quarterly.

Compounded quarterly (n = 4)?

Exponential growth

( )toy A r

(1 )ty P r

1. The number of bacteria of a certain type in a certain person’s body doubles every 40 minutes. If 6 are present at 2 A.M., how many will there be at 12 noon?

0

15

( )

6(2)

6(32,768)

196,608

ty A r

y

y

y

2. In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of cell subscribers increased by 75 % per year after 1985. How many cell phone subscribers in Centerville last year 2000?

Answer: 1, 260, 131 (rounded value)

EXPONENTIAL DECAY

• When an antibiotic is added to the culture of bacteria, the number of bacteria is reduced by half every 3 hours. If there are 80, 000 bacteria, how many bacteria are left after a day?

0

81

80,0002

312.5

313

xy A r

y

y

y

• A radioactive substance decays so that every year 75% of the previous year’s substance remains. If there are 4 kg of substance at the start of the experiment, how much remains after 3 years?

t

3

3

y = P(1 - r)

= 4 (1 - 0.75)

= 4 ( 0.25)

= 4(0.0156)

= 0.0625 kg

• The number of wolves in the wild in the northern section of a certain country is decreasing at the rate of 3.5 % per year. Your environmental studies class as counted 80 wolves in the area. How many wolves are left after 10 years?

10

1

80(1 0.035)

56.02

56

ty P r

y

y

y

EXERCISES

1. A Php 1,000-deposit is made at a bank that pays 12% interest compounded annually. How much will you have in your account at the end of 10 years?

Given:P = Php 1, 000r = 12 %t = 10 years n = 1Find: Deposited amount after 10 years

Answer: Php 3, 105.85

2. A Php 1,000 deposit is made at a bank that pays 12% interest compounded quarterly. How much will you have in your account at the end of 10 years?

Given:P = Php 1, 000r = 12 % compounded quarterlyt = 10 years Find: Deposited amount after 10 years

Answer: Php 3,262.04

3. Suppose Mitch has Php 1, 000 that she invests in an account that pays 3.5% interest compounded quarterly. How much money does she have at the end of 5 years?

Given:P = Php 1, 000r = 3.5 % compounded quarterlyt = 5 years Find: Deposited amount after 5 years

Answer: Php 1,190.34

4. How much will be on your account after 5 years if you deposited Php 10,000 at 20 % per annum compounded semi annually?

Answer: Php 25, 937. 42

• A population of 200 frogs increases at an annual rate of 22%. How many frogs will there be in 10 years?

Answer: Php 1, 461

• Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. How many players will remain after 5 rounds?

Answer: 4 players

• Suppose the amount in grams of plutonium 241 present in a given sample is determined by the function defined by .

• where t is measured in years. Find the amount present in the sample after the 4 years.

0.053( ) 2.00 tA t e

0.053

0.053(4)

0.212

( ) 2.00

2

2

1.6

tA t e

e

e

grams