Exponential model word problem: medication dissolve | High School Math | Khan Academy

Name: Exponential model word problem: medication dissolve | High School Math | Khan Academy
Uploaded: 2016-02-18T18:29:55.000Z
Duration: 4 min 4 s
Channel: Khan Academy
Description: - The relationship between the elapsed time and the amount of medication in Carlos' bloodstream is given by the function M(t) = 20e^(-0.8t), where t is the time in hours and M(t) is the amount of medication in milligrams. - To find the time when Carlos will have 1 milligram of medication remaining,

February 18, 2016

Khan Academy

TL;DR

The video explains how to calculate the time it takes for a medication to deplete in the bloodstream.

Transcript

[Voiceover] Carlos has taken an initial dose of a prescription medication. The relationship between the time, between the elapsed time t, in hours, since he took the first dose and the amount of medication, M of t, in milligrams, in his bloodstream is modeled by the following function. Alright, in how many hours will Carlos have 1 milligram of me... Read More

Key Insights

⌛ The relationship between time and the amount of medication can be modeled using exponential functions.
❓ Natural logarithms can be used to solve equations involving exponential functions.
✊ Taking the natural logarithm of a number represents the power to which the base must be raised to obtain that number.
⌛ The rounded value for time indicates the time it takes for the medication dosage to deplete to a specific value.

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Questions & Answers

Q: How is the relationship between time and the amount of medication in Carlos' bloodstream modeled?

The relationship is modeled by the function M(t) = 20e^(-0.8t), where t is the time in hours and M(t) is the amount of medication in milligrams.

Q: What is the equation we need to solve to find the time when Carlos will have 1 milligram of medication remaining?

We need to solve the equation M(t) = 1, which can be written as 20e^(-0.8t) = 1.

Q: How do we solve the equation to find the value of t?

By taking the natural logarithm of both sides, we simplify the equation to -0.8t = ln(0.05). Dividing both sides by -0.8 gives us t ≈ 3.74 hours.

Q: What is the significance of the rounded value for t?

The rounded value, 3.74 hours, represents the time it takes for Carlos' medication dosage to go down to 1 milligram from the initial 20 milligrams.

Summary & Key Takeaways

The relationship between the elapsed time and the amount of medication in Carlos' bloodstream is given by the function M(t) = 20e^(-0.8t), where t is the time in hours and M(t) is the amount of medication in milligrams.
To find the time when Carlos will have 1 milligram of medication remaining, we need to solve M(t) = 1.
Taking the natural logarithm of both sides and solving the equation gives t = 3.74 hours.

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Transcript

[Voiceover] Carlos has taken an initial dose of a prescription medication. The relationship between the time, between the elapsed time t, in hours, since he took the first dose and the amount of medication, M of t, in milligrams, in his bloodstream is modeled by the following function. Alright, in how many hours will Carlos have 1 milligram of me... Read More

Key Insights

⌛ The relationship between time and the amount of medication can be modeled using exponential functions.

❓ Natural logarithms can be used to solve equations involving exponential functions.

✊ Taking the natural logarithm of a number represents the power to which the base must be raised to obtain that number.

⌛ The rounded value for time indicates the time it takes for the medication dosage to deplete to a specific value.

Questions & Answers

Q: How is the relationship between time and the amount of medication in Carlos' bloodstream modeled?

The relationship is modeled by the function M(t) = 20e^(-0.8t), where t is the time in hours and M(t) is the amount of medication in milligrams.

Q: What is the equation we need to solve to find the time when Carlos will have 1 milligram of medication remaining?

We need to solve the equation M(t) = 1, which can be written as 20e^(-0.8t) = 1.

Q: How do we solve the equation to find the value of t?

By taking the natural logarithm of both sides, we simplify the equation to -0.8t = ln(0.05). Dividing both sides by -0.8 gives us t ≈ 3.74 hours.

Q: What is the significance of the rounded value for t?

The rounded value, 3.74 hours, represents the time it takes for Carlos' medication dosage to go down to 1 milligram from the initial 20 milligrams.

Summary & Key Takeaways

The relationship between the elapsed time and the amount of medication in Carlos' bloodstream is given by the function M(t) = 20e^(-0.8t), where t is the time in hours and M(t) is the amount of medication in milligrams.

To find the time when Carlos will have 1 milligram of medication remaining, we need to solve M(t) = 1.

Taking the natural logarithm of both sides and solving the equation gives t = 3.74 hours.