   Chapter 4.1, Problem 70E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function C ( t ) = 8 ( e − 0.4 t − e − 0.6 t ) where the time t is measured in hours and C is measured in μg/mL. What is the maximum concentration of the antibiotic during the first 12 hours?

To determine

To find: The maximum concentration of the antibiotic for the first 12 hours after an antibiotic tablet is taken.

Explanation

Given:

The concentration of the antibiotic in the blood stream is given by the function c(t)=8(e0.4te0.6t), where time t is measured in hours and c is measured in μgmL.

Calculation:

Since the time starts after consuming the antibiotic tablet, consider the initial stage as zero.

The process duration is 12 hours.

Therefore, the process interval will be [0,12].

Obtain the first derivative of the function c(t).

c(t)=ddt(8(e0.4te0.6t))=8((0.4)e0.4t(0.6)e0.6t)=8(0.6e0.6t0.4e0.4t)

Set c(t)=0 and obtain the critical numbers.

8(0.6e0.6t0.4e0.4t)=00.6e0.6t0.4e0.4t=00.60.4=e0.4te0.6t32=e0.4te0.6t

Apply natural logarithm on both sides,

ln(32)=ln(e0

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