5.2

Copyright © 2014 Pearson Education, Inc.

Applications of the Models

OBJECTIVE• Perform computations involving interest compounded continuously and continuous money flow.

• Calculate the total consumption of a natural resource.

• Find the present value of an investment.

Slide 5- 2Copyright © 2014 Pearson Education, Inc.

5.2 Applications of the Models

Growth Formula: 000

1 .T kt kTP

P e dt ek

Decay Formula: 000

1 .T kt kTP

P dt ek

Slide 5- 3Copyright © 2014 Pearson Education, Inc.

5.2 Applications of the Models

Definition

If is invested for t years at interest rate k, compounded continuously, then

where The value of P is called the future value of dollars invested at interest rate k, compounded continuously, for t years.

0P

0 ,ktP t P e

0 at 0.P P t 0P

Slide 5- 4Copyright © 2014 Pearson Education, Inc.

5.2 Applications of the Models

Quick Check 1

Find the future value of $10,000 invested for 3 years at an interest rate of 6% compounded continuously.

We know that So, 0 0, $10,000, 3, and 0.06.ktP t P e P t k

0.06 33 10,000P e 0.1810,000e 10,000 1.197217

$11,972.17

Slide 5- 5Copyright © 2014 Pearson Education, Inc.

FUTURE VALUE OF A CONTINUOUS MONEY FLOW

If the yearly flow of money into an investment is given by some function R(t), then the future value of the continuous money flow at interest rate k, compounded continuously over T years, is given by

0

.T ktA R t e dt

5.2 Applications of the Models

Slide 5- 6Copyright © 2014 Pearson Education, Inc.

Example 1: Find the future value of the continuous money flow if $1000 per year flows at a constant rate into an account paying 8%, compounded continuously, for 15 yr.

5.2 Applications of the Models

15

15 0.08 0.08

00

1000$1000

0.08t te dt e

0.08 15 0.08 01000

0.08e e

1.210001 $29,001.46

0.08e

Slide 5- 7Copyright © 2014 Pearson Education, Inc.

Example 2: Consider a continuous flow of money into an investment at the constant rate of P0 dollars per

year. What should P0 be so that the amount of a

continuous money flow over 20 yr, at an interest rate of 8%, compounded continuously, will be $10,000?

5.2 Applications of the Models

200.08

00

110,000

0.08tP e

20 0.08

0010,000 tP e dt

Slide 5- 8Copyright © 2014 Pearson Education, Inc.

Example 2 (concluded):

A continuous money flow of $202.38 per year, invested at 8%, compounded continuously for 20 years, will yield $10,000.

5.2 Applications of the Models

0$202.38 P

010,000 49.4129053P

1.6

010,000 12.5 12.5P

0.08 20

010,000 12.5 12.5P e

Slide 5- 9Copyright © 2014 Pearson Education, Inc.

DEFINITION:

The present value, P0, of an amount P, when P0 is

invested at interest rate k, compounded continuously, and due t years later, is given by

0

0

kt

kt

P P e

PP

e

5.2 Applications of the Models

0ktP Pe

Slide 5- 10Copyright © 2014 Pearson Education, Inc.

Example 3: Find the present value of $200,000 due 25 yr from now, at 8.7% compounded continuously.

P0 200,000e 0.087(25) $22,721.63

Thus the present value is $22,721.63.

5.2 Applications of the Models

Slide 5- 11Copyright © 2014 Pearson Education, Inc.

5.2 Applications of the Models

Quick Check 2

Mira Bell, following the birth of a grandchild, wants to set up a trust fund that will be worth $120,000 on the child’s 18th birthday. Mira can get an interest rate of 5.6%, compounded continuously for the time period. What amount will Mira have to deposit in the trust fund to achieve her goal?

0ktP Pe

0.056 180 120,000P e

1.0080 120,000P e

0 120,000 0.364948146P

0 43,793.78P

So Mira must deposit $43,793.78 into the trust fund to achieve her goal.

Slide 5- 12Copyright © 2014 Pearson Education, Inc.

DEFINITION:

The accumulated present value, B, of a continuous money flow into an investment at a rate of R(t) dollars per year from now until T years in the future is given by

where k is the interest rate, and interest is compounded continuously.

( ) ,T kt

oB R t e dt

5.2 Applications of the Models

Slide 5- 13Copyright © 2014 Pearson Education, Inc.

Example 4: Find the accumulated present value of an investment over a 5-yr period if there is a continuous money flow of $2400 per year and the interest rate is 14%, compounded continuously.

5.2 Applications of the Models

5 0.14

02400 te dt

50.14

0

2400

0.14te

0.14 5 0.14 017,142.86 e e

0.717,142.86 1e

$8629.97

Slide 5- 14Copyright © 2014 Pearson Education, Inc.

Consumption of a Natural ResourceSuppose that P(t) is the amount of a natural resource used at time t. If consumption of the resource is growing exponentially at growth rate k, then the total amount used during the interval [0, T ] is given by

where P0 represents the amount of the natural resource used at time t = 0.

P0ekt

0

T

dt P0

kekT 1 ,

5.2 Applications of the Models

Slide 5- 15Copyright © 2014 Pearson Education, Inc.

The area under the graph of

over the interval [0, T] is is given by

000

1 .T kt kTP

P e dt ek

P t P0ekt

5.2 Applications of the Models

Slide 5- 16Copyright © 2014 Pearson Education, Inc.

Example 5: In 2000 (t = 0), world gold production was 2547 metric tons, and it was growing exponentially at the rate of 0.6% per year. If the growth continues at this rate, how many tons of gold will be produced from 2000 to 2013?

5.2 Applications of the Models

34,437 metric tons

0.078424,500 1e

13 0.006

02547 te dt 0.006 13 0.006 02547

0.006e e

Slide 5- 17Copyright © 2014 Pearson Education, Inc.

Example 6: The world reserves of gold in 2000 were estimated to be 77,000 metric tons. Assuming that the growth rate for production given in Example 5 continues and that no new reserves are discovered, when will the world reserves of gold be depleted?

5.2 Applications of the Models

Slide 5- 18Copyright © 2014 Pearson Education, Inc.

Example 6 (concluded):

5.2 Applications of the Models

77,000 0.00625471

0.006Te

77,000 0.006424,500 1Te

0.1814 0.006 1Te

1.1814 0.006Te

ln1.1814 0.006ln Te

ln1.1814 0.006T

28 yrs T

Slide 5- 19Copyright © 2014 Pearson Education, Inc.

5.2 Applications of the Models

Quick Check 3

The movie Avatar is set in the year 2154 on the moon Pandora, of the planet Polyphemus in the star system of Alpha Centauri. The conflict in the movie is centered around a precious but scarce mineral, Unobtanium.

a.) In 2010, the universe’s production of Unobtanium was 6800 metric tons and it was being used at a rate of 0.8% per year. If Unobtanium continues to be used at this rate, how many tons of Unobtanium will be used between 2010 and 2024?

b.) In 2010, the universe’s reserve of Unobtanium was 86,000 metric tons. Assuming that the growth rate of 0.8% per year continues and that no new reserves are discovered, when will the universe reserves of Unobtanium be depleted?

Slide 5- 20Copyright © 2014 Pearson Education, Inc.

5.2 Applications of the Models

Quick Check 3 Continued

a.) In 2010, the universe’s production of Unobtanium was 6800 metric tons and it was being used at a rate of 0.8% per year.If Unobtanium continues to be used at this rate, how many tons of Unobtanium will be used between 2010 and 2024?

0 1kTPP e

k 0.008 146800

10.008

e

0.112850,000 1e

850,000 .11851286

100,736So 100,736 metric tons of Unobtanium will be used between 2010 and 2024.

Slide 5- 21Copyright © 2014 Pearson Education, Inc.

5.2 Applications of the Models

Quick Check 3 Concluded

b.) In 2010, the universe’s reserve of Unobtanium was 86,000 metric tons. Assuming that the growth rate of 0.8% per year continues and that no new reserves are discovered, when will the universe reserves of Unobtanium be depleted?

0.008680086000 1

0.008Te

0 1kTPP e

k

0.11286000 850000 1e 0.0081.1012 Te

ln1.1012 0.008 T12 T

So the universe reserves of Unobtanium will be depleted in 12 years, or by the year 2022.

Slide 5- 22Copyright © 2014 Pearson Education, Inc.

5.2 Applications of the Models

Section Summary

• The future value of an investment is given by where dollars are invested for t years at interest rate k, compounded continuously.

• The accumulated future value of a continuous income stream is given by

where represents the rate of the continuous income stream, k is the interest rate, compounded continuously, at which the continuous income stream is invested, and T is the number of years for which the income stream is invested.

0 ,ktP P e 0P

0

,T ktA R t e dt

R t

Slide 5- 23Copyright © 2014 Pearson Education, Inc.

5.2 Applications of the Models

Section Summary Continued

• If is a constant function, then

• The present value is given by where the amount P is due t years later is invested at interest rate k, compounded continuously.

R t

1 .kTR tA e

k

0 ,ktP Pe

Slide 5- 24Copyright © 2014 Pearson Education, Inc.

5.2 Applications of the Models

Section Summary Concluded

• The accumulated present value of a continuous income stream is given by

where represents the rate of the continuous income stream, k is the interest rate, compounded continuously, at which the continuous income stream is invested, and T is the number of years over which the income stream is received.

• If is a constant function, then

0

,T ktB R t e dt

R t

R t

1 .kTR tB e

k