Kevin M.

asked • 08/24/21# Caffeine Problems

Caffeine Problems - Exponents

Directions: You are contemplating the installation of small coffee shops spaced intermittently throughout the park. Before you choose the locations and frequency of the shops, you want to study how caffeine reacts in the body.

The half-life of caffeine is 5 hours; this means that approximately 1/2 of the caffeine in the bloodstream is eliminated every 5 hours. Suppose you drink a 16-ounce drink that contains 80 mg of caffeine. Suppose the caffeine in your bloodstream peaks at 80 mg.

How much caffeine will remain in your bloodstream after 5 hours? 10 hours? 1 hour? 2 hours? Record your answers in the table. Explain how you came up with your answers. (You can return to your answers later to make any corrections if you find your strategy was incorrect.)

Time (hours) since peak level reached | 0 | 1 | 2 | 5 | 10 |

Caffeine in bloodstream (mg) | 80 |

Write an exponential function f to model the amount of caffeine remaining in the bloodstream *t* hours after the peak level. What does your exponent need to represent? How can you determine this exponent if you know the number of hours that have passed?

Use the function you wrote in #2 to check your answers for the table in #1. Make any necessary corrections. (Be careful when entering fractional exponents in the calculator. Use parenthesis!)

Time (hours) since peak level reached | 0 | 1 | 2 | 5 | 10 |

Caffeine in bloodstream (mg) | 80 |

Determine the amount of caffeine remaining in the bloodstream:

3 hours after the peak level?

What about 8 hours after peak level?

20 hours?

The half-life of caffeine varies among individuals. For example, some medications extend the half-life to 8 hours. This means that 1/2 of the caffeine is eliminated from the bloodstream every 8 hours. Write a function for this new half-life time (assuming a peak level of 80 mg of caffeine)

Determine the amount of caffeine in the bloodstream after:

1 hour?

5 hours?

10 hours?

20 hour?

## 1 Expert Answer

Hello, Kevin,

The formula for half-life calculations is f(t) = N*(1/2)^{t/5}, where t is the time in hours, N is the initial value (concentration in this case) and 5 is the half-life in hours. That formula leads to this table:

A half life of 5 years would lead to 1/2 of the caffeine disappearing. 10 years it would be down 1/2 again. These numbers reflect that prediction.

Use this same approach for the second half of the equation, using 8 hours instead of 5.

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Mark M.

What is preventing you from following these explicit instructions?08/24/21