# The maximum concentration of the drug. ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 3.1, Problem 68E
To determine

## To evaluate: The maximum concentration of the drug.

Expert Solution

The maximum concentration of the drug is 4.5mg/L .

### Explanation of Solution

Given:

The concentration of the drug in the patient bloodstream after t minute is given by the function C(t)=0.06t0.0002t2 , where 0t240 .

Calculation:

The function of the concentration of the drugs in the patient bloodstream is given by,

C(t)=0.06t0.0002t2=0.0002t2+0.06t (1)

The standard form of function,

f(t)=at2+bt+c (2)

The maximum or minimum value of the function occurs at,

t=b2a (3)

If a>0 , then the minimum value is f(b2a) .

If a<0 , then the maximum value is f(b2a) .

From the equation (1) and (2),

a=0.0002b=0.06

Substitute 0.0002 for a and 0.06 for b in the equation (3),

t=0.062×0.0002=0.060.0004=6004=150

The time taken to reach maximum concentration is 150minutes .

The function has maximum value,

a=0.0002<0 .

Substitute 150 for t in the equation (1) for the maximum concentration,

C(150)=0.06×1500.0002×(150)2=94.5=4.5

The maximum concentration of the drugs after 150 minutes is 4.5mg/L .

Thus, the maximum concentration of the drug is 4.5mg/L .

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