1. Homework Homework Assignment #9 Review Section 5.8
Page 365, Exercises: 1 – 49(EOO)
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Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 366
1. A bacteria population P obeys the exponential growth law P(t) = 2,000e1.3t (t in hours). (a) How many bacteria are present initially? (b) At what time will there be 10,000 bacteria?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

3. Homework, Page 366
5. The decay constant of Cobalt–60 is 0.13 years–1. What is its half-life?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 366
9. Find the solution to y ′ = 3y satisfying y(2) = 4.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 366
13. Assuming that population growth is approximately exponential, which of the two sets of data is most likely to represent the population (in millions) of a city over a 5-year period?
Year
2000
2001
2002
2003
2004
Data I
3.14
3.36
3.60
3.85
4.11
Data II
3.24
3.54
4.04
4.74
Data II is most likely to represent the population of the city over the
five-year period, as Data I is almost linear.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 366
17. A 10-kg quantity of radioactive isotope decays to 3-kg after 17 years. Find the decay constant of the isotope.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

7. Homework, Page 366
21. The atmospheric pressure P(h) (in psi) at a height h (in miles) above sea level on earth satisfies a differential equation P′ = – kP for some positive constant k. (a) Measurements with a barometer show that P(0) = 14.7 and P(10) = What is the decay constant k?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

8. Homework, Page 366
21. (b) Determine the atmospheric pressure 15 miles above sea level.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

9. Homework, Page 366
25. In 1965, Gordon Moore predicted that the number of transistors on a microchip would increase exponentially. (a) Does the table of data confirm Moore’s prediction? If so, estimate the growth constant.
Year
No. Trans,
1971
2,250
1972
2,500
1974
5,000
1978
29,000
1982
120,000
1985
275,000
1989
1,180,000
1993
3,100,000
1997
7,500,000
1999
24,000,000
2000
42,000,000
The data seems to support
Moore’s prediction.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

10. Homework, Page 366 25. (b) Plot the data in the table. Year No. Trans,
1971
2,250
1972
2,500
1974
5,000
1978
29,000
1982
120,000
1985
275,000
1989
1,180,000
1993
3,100,000
1997
7,500,000
1999
24,000,000
2000
42,000,000
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

11. Homework, Page 366
25. (c) Let N(t) be the number of transistors t years after Find an approximate formula N(t) ≈ Cekt, where t is the number of years after 1971.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

12. Homework, Page 366
25. (d) Estimate the doubling time in Moore’s Law for the period 1971 – (e) If Moore’s Law holds to the end of the decade, how many transistors will a microchip hold in 2010?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

13. Homework, Page 366
25. (e) Can Moore expect his prediction to hold indefinitely? Moore cannot expect his prediction to hold indefinitely, as a some point transistors will get as small as they can, one or more molecules, in size. That coupled with the finite length of the connecting conductors will limit the miniaturization we have seen over the last several decades.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

14. Homework, Page 366
29. A certain quantity increases quadratically: P(t) =P0t2. (a) Starting at time t0 = 1, how long will it take for P to double in size? How long will it take starting at t0 = 2 or 3? (b) In general, starting at time t0, how long will it take for P to double in size?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

15. Homework, Page 366
33. A bank pays interest at the rate of 5%. What is the yearly multiplier, if interest is compounded (a) annually? 1.05 (b) three times per year? (c) continuously?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

16. Homework, Page 366
37. An investment increases in value at a continuously compounded rate of 9%. How large must the initial investment be to build up a value of $50,000 over a seven-year period?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

17. Homework, Page 366
41. If a company invests $2,000,000 to upgrade a factory, it will earn additional profits of $500,000 per year. Is the investment worthwhile, assuming an interest rate of 6%?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

18. Homework, Page 366
45. Use equation 3 to compute PV of an income stream paying out R(t) = $5,000/yr continuously for ten yr at r = 0.05.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

19. Homework, Page 366
49. Show that PV of an investment that pays out R dollars/yr continuously for T years is R(1 – e–rt)/r where r is the interest rate.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

20. Homework Homework Assignment #10 Review Sections 5.1 – 5.8
Page 369, Exercises: 1 – 97(EOO)
Quiz next time
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company