2.5.2: Exercise M.5

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The half-life of a drug in the blood is the amount of time it takes for half the drug to be eliminated from the body. This amount of time is fixed per individual and is independent of the initial amount of the drug. So, for example, if the half-life of a certain drug is 3 hours, and 25 mg of the drug is in the blood at 5 p.m., then 12.5 mg (half of 25 mg) of the drug is left at 8 p.m., and 6.25 mg (half of 12.5 mg) is left at 11 p.m.

Recall Question 4 from the collaboration: “The rate of elimination of caffeine from the human body varies greatly from individual to individual. Suppose that Jacob drinks a 16-ounce coffee drink that contains about 310 mg of caffeine. His body eliminates about 13% of the caffeine every hour.” You created an algebraic model for this situation. Use your algebraic model to help you answer the questions below. (Hint: You may find it helpful to make a table (or use a spreadsheet) with two columns, one labeled “Time Since Drinking Coffee” and the second labeled “Amount of Caffeine in Body.”)

(a) Approximately how long will it take for Jacob’s body to eliminate half of the caffeine? (As mentioned above, this amount of time is called the caffeine’s “half-life.”) Write your answer rounded to the nearest whole hour.

(b) If Jacob drank a 12-ounce Pepsi containing 38 mg of caffeine instead of the coffee drink, how long will it take for Jacob’s body to eliminate half of the caffeine? Write your answer rounded to the nearest whole hour.

(2) In Question 1 above, we assumed that the body could eliminate 13% of the caffeine present every hour. Wikipedia states that the half-life of caffeine in a healthy adult is often between 4.9 and 6 hours.²⁴ (Remember, the “half-life” refers to the amount of time it takes for the body to eliminate half of the caffeine currently present in the body.)

If 13% of the caffeine is eliminated every hour, does this imply a half-life between 4.9 and 6 hours? Explain. Does this answer agree with your answer to Question 1(a) above?

(3) The antibiotic tetracycline is used for treatment against many different bacterial infections. Suppose that an individual is given 300 mg of tetracycline every 6 hours, and the half-life of tetracycline is 8 hours. Round to the nearest mg for all answers below. (Hint: Assume all tetracycline is absorbed into the body immediately.)

(a) If the individual takes his first dose at noon, how much tetracycline remains in his body just before he takes his second dose?

(b) How much tetracycline is in the individual’s body just right after he takes his second dose?

(4) Joan quit her job at GM when she was 35 years old. Her friend, who is a financial advisor, recommended that she leave her 401(k) retirement savings in GM’s retirement plan rather than withdrawing or transferring the money to a new plan. Her friend said that, on average, Joan could expect a 6% increase per year if she left her money in the plan for many years, based on the past performance of GM’s plan. Joan will not be able to add more money to the GM 401(k) account, but can open a new one at her new job.

Create a model for the amount of money Joan will have in her 401(k) after any number of years. Use A for the accumulated amount of money, t for the number of years, and P for the amount of money in the plan when she quit, which is called the principal.

What percent of the principal will Joan have if she leaves the money in the account for the 32 years until she retires? What percent increase does this represent? Round each answer to the nearest hundredth of a percent.

If Joan’s original principal was $20,100, how much would be in the account when she retires? Round to the nearest cent.

Approximately how many years will it take the principal to double? How long to quadruple? Write your answers rounded to the nearest year.

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²⁴ http://en.Wikipedia.org/wiki/Caffeine

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