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In this lesson, we will learn how to use the exponential decay model in applications including radioactive decay, Newtonβs law of cooling, and the concept of half-life.

Q1:

A doctor injected a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there were 4.75 milligrams of dye remaining in the patientβs system. Which of the following is an appropriate model for this situation?

Q2:

The rate at which a radioactive substance decays is proportional to the remaining number of atoms. The differential equation which can be used to describe this process follows where π represents the number of atoms remaining after π‘ seconds. The proportionality constant π is considered the decay constant for this process. If π ο¦ represents the number of remaining atoms at π‘ = 0 seconds, find the general solution.

Q3:

The results of a medical study showed that, in healthy adults, the half-life of caffeine is 5.7 hours. So, if an adult consumes 250 mg of caffeine in their breakfast coffee at 6 am, they will have approximately 125 mg of caffeine in their system at 11:40 am.

If a person drinks a can of cola containing 30 mg of caffeine, the amount of caffeine, πΆ , in their system π‘ hours later can be found using the equation πΆ = 3 0 οΌ 1 2 ο ( ) π‘ 5 . 7 .

Write the equation in the form πΆ = π΄ ( π ) π‘ , giving values to 3 decimal places if necessary.

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