1. 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.

2. Slide 5- 2 Copyright © 2014 Pearson Education, Inc. 5.2 Applications of the Models Growth Formula: Decay Formula:

3. Slide 5- 3 Copyright © 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.

4. Slide 5- 4 Copyright © 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,

5. Slide 5- 5 Copyright © 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 5.2 Applications of the Models

6. Slide 5- 6 Copyright © 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

7. Slide 5- 7 Copyright © 2014 Pearson Education, Inc. Example 2: Consider a continuous flow of money into an investment at the constant rate of P 0 dollars per year. What should P 0 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

8. Slide 5- 8 Copyright © 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

9. Slide 5- 9 Copyright © 2014 Pearson Education, Inc. DEFINITION: The present value, P 0, of an amount P, when P 0 is invested at interest rate k, compounded continuously, and due t years later, is given by 5.2 Applications of the Models

10. Slide 5- 10 Copyright © 2014 Pearson Education, Inc. Example 3: Find the present value of $200,000 due 25 yr from now, at 8.7% compounded continuously. Thus the present value is $22,721.63. 5.2 Applications of the Models

11. Slide 5- 11 Copyright © 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 18 th 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? So Mira must deposit $43,793.78 into the trust fund to achieve her goal.

12. Slide 5- 12 Copyright © 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. 5.2 Applications of the Models

13. Slide 5- 13 Copyright © 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

14. Slide 5- 14 Copyright © 2014 Pearson Education, Inc. Consumption of a Natural Resource Suppose 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 P 0 represents the amount of the natural resource used at time t = 0. 5.2 Applications of the Models

15. Slide 5- 15 Copyright © 2014 Pearson Education, Inc. The area under the graph of over the interval [0, T] is is given by 5.2 Applications of the Models

16. Slide 5- 16 Copyright © 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

17. Slide 5- 17 Copyright © 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

18. Slide 5- 18 Copyright © 2014 Pearson Education, Inc. Example 6 (concluded): 5.2 Applications of the Models

19. Slide 5- 19 Copyright © 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?

20. Slide 5- 20 Copyright © 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? So 100,736 metric tons of Unobtanium will be used between 2010 and 2024.

21. Slide 5- 21 Copyright © 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? So the universe reserves of Unobtanium will be depleted in 12 years, or by the year 2022.

22. Slide 5- 22 Copyright © 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.

23. Slide 5- 23 Copyright © 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.

24. Slide 5- 24 Copyright © 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