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## With what kind of exponential model would half-life be associated? What role does half-life play in these models?

sectioN exercises 549

6.7 SeCTIOn exeRCISeS

veRbAl 1. With what kind of exponential model would half-life

be associated? What role does half-life play in these models?

2. What is carbon dating? Why does it work? Give an example in which carbon dating would be useful.

3. With what kind of exponential model would doubling time be associated? What role does doubling time play in these models?

4. Define Newton’s Law of Cooling. Then name at least three real-world situations where Newton’s Law of Cooling would be applied.

5. What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain.

nUmeRIC

6. The temperature of an object in degrees Fahrenheit after t minutes is represented by the equation T(t) = 68e −0.0174t + 72. To the nearest degree, what is the temperature of the object after one and a half hours?

For the following exercises, use the logistic growth model f (x) = 150 _ 1 + 8e−2x

.

7. Find and interpret f (0). Round to the nearest tenth. 8. Find and interpret f (4). Round to the nearest tenth. 9. Find the carrying capacity. 10. Graph the model.

11. Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data.

12. x –2 –1 0 1 2 3 4 5

f (x) 0.694 0.833 1 1.2 1.44 1.728 2.074 2.488

13. Rewrite f (x) = 1.68(0.65)x as an exponential equation with base e to five significant digits.

TeChnOlOgy For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.

14. x 1 2 3 4 5 6 7 8 9 10

f (x) 2 4.079 5.296 6.159 6.828 7.375 7.838 8.238 8.592 8.908

15. x 1 2 3 4 5 6 7 8 9 10

f (x) 2.4 2.88 3.456 4.147 4.977 5.972 7.166 8.6 10.32 12.383 16. x 4 5 6 7 8 9 10 11 12 13

f (x) 9.429 9.972 10.415 10.79 11.115 11.401 11.657 11.889 12.101 12.295 17.

x 1.25 2.25 3.56 4.2 5.65 6.75 7.25 8.6 9.25 10.5

f (x) 5.75 8.75 12.68 14.6 18.95 22.25 23.75 27.8 29.75 33.5

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in t years is modeled by the equation P(t) = 1000 _

1 + 9e−0.6t .

18. Graph the function. 19. What is the initial population of fish? 20. To the nearest tenth, what is the doubling time for

the fish population? 21. To the nearest whole number, what will the fish

population be after 2 years? 22. To the nearest tenth, how long will it take for the

population to reach 900? 23. What is the carrying capacity for the fish population?

Justify your answer using the graph of P.

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550 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

exTenSIOnS 24. A substance has a half-life of 2.045 minutes. If the

initial amount of the substance was 132.8 grams, how many half-lives will have passed before the substance decays to 8.3 grams? What is the total time of decay?

25. The formula for an increasing population is given by P(t) = P0e

rt where P0 is the initial population and r > 0. Derive a general formula for the time t it takes for the population to increase by a factor of M.

26. Recall the formula for calculating the magnitude of an earthquake, M = 2 _ 3 log S __ S0 . Show each step for solving this equation algebraically for the seismic moment S.

27. What is the y-intercept of the logistic growth model y = c ________ 1 + ae−rx ? Show the steps for calculation. What does this point tell us about the population?

28. Prove that b x = e xln(b) for positive b ≠ 1.

ReAl-WORld APPlICATIOnS For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.

29. To the nearest hour, what is the half-life of the drug? 30. Write an exponential model representing the amount of the drug remaining in the patient’s system after t hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after 3 hours. Round to the nearest milligram.

31. Using the model found in the previous exercise, find f (10) and interpret the result. Round to the nearest hundredth.

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of 1.15% per day.

32. To the nearest day, how long will it take for half of the Iodine-125 to decay?

33. Write an exponential model representing the amount of Iodine-125 remaining in the tumor after t days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram.

34. A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?

35. The half-life of Radium-226 is 1590 years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

36. The half-life of Erbium-165 is 10.4 hours. What is the hourly decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

37. A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is 5730 years.)

38. A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?