Question

Immediately following an injection, the concentration of a drug in the bloodstream is $ 300 $ milligrams per milliliter. After $ t $ hours, the concentration is $ 75\% $ of the level of the previous hour. (a) Find a model for $ C(t) $, the concentration of the drug after $ t $ hours. (b) Determine the concentration of the drug after $ 8 $ hours.

Name: Problem 76, Chapter 3: Precalculus with Limits
Uploaded: 2020-06-16T20:10:51-08:00
Duration: 49 s
Channel: Heather Zimmers
Description: Immediately following an injection, the concentration of a drug in the bloodstream is 300 milligrams per milliliter. After t hours, the concentration is 75% of the level of the previous hour. (a) Find a model for C(t) , the concentration of the drug after t hours. (b) Determine the concentration of the drug after 8 hours.

   Immediately  following  an injection,  the  concentration  of  a  drug  in  the  bloodstream is $ 300 $ milligrams  per  milliliter.  After $ t $ hours,  the  concentration  is $ 75\% $ of  the  level  of  the  previous  hour.

(a)  Find  a  model  for $ C(t) $, the  concentration  of  the  drug after $ t $ hours.
(b)  Determine  the  concentration  of  the  drug  after $ 8 $ hours.

Precalculus with Limits

Ron Larson 2nd Edition

Chapter 3, Problem 76

Instant Answer

Solved by Expert Heather Zimmers

06/16/2020

Step 1

This can be modeled by an exponential decay function, where the base is 0.75 (representing the 75% decrease) and the exponent is $t$ (representing the time in hours). The initial concentration of the drug is represented by the coefficient 300. Therefore, the model Show more…

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Immediately following an injection, the concentration of a drug in the bloodstream is $ 300 $ milligrams per milliliter. After $ t $ hours, the concentration is $ 75\% $ of the level of the previous hour. (a) Find a model for $ C(t) $, the concentration of the drug after $ t $ hours. (b) Determine the concentration of the drug after $ 8 $ hours.

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Key Concepts

Exponential Function Modeling

Exponential function modeling uses the formula C(t) = C? * b^t to represent quantities that change multiplicatively over time. In these models, C? is the initial amount, and b is the decay factor, representing the fraction of the quantity that remains after each time unit. This method is widely applicable in fields like pharmacokinetics to predict drug concentration levels at different times.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value over equal time intervals. This concept is fundamental when modeling situations where a substance or quantity, such as a drug concentration, diminishes over time according to a consistent percentage decrease per time unit.

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