   Chapter 2.3, Problem 58E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Alcohol After a large number of drinks, a person has a blood alcohol level of 200 mg/dL (milligrams per deciliter). If the amount of alcohol in the blood decays exponentially, with one fourth being removed every hour, find the time it will take for the person's blood alcohol level to decrease to 80mg/dL.

To determine

To calculate: The time it will take the level of alcohol to decrease to 80 mg/dL if initially patient has absorbed 200 mg of the alcohol and one by fourth of the remaining alcohol is being removed every hours.

Explanation

Given Information:

The alcohol in the blood decays exponentially. One by fourth of the remaining alcohol is being removed after every hours. The patient has alcohol level 200 mg/dL in the blood initially. The final alcohol level is 80 mg/dL.

Formula used:

The expression of exponential decay model is,

Q(t)=Q0ekt

Here, Q(t) is remaining amount at time t, Q0 is initial amount of the sample, k is decay constant and t is time in year.

The relation between decay constant and half-life time is,

tdk=ln2

Here, td is half-life time.

Calculation:

Consider the expression of exponential decay model

Q(t)=Q0ekt

Since One by fourth of the remaining alcohol is being removed after every hours. Therefore after one hour reaming alcohol contain in the patient is,

Q=Q0(114)=34Q0

Therefore substitute t=1 and Q=34Q0 in the expression Q(t)=Q0ekt.

34Q0=Q0ek(1)

Divide both side by Q0

34Q0Q0=Q0ekQ034=ek

Take natural log on both side of the equation

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