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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

The average blood alcohol concentration (BAC) of eight male subjects was measured after consumption of 15 mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration function

C(t) = 0.0225te–0.0467t

where t is measured in minutes after consumption and C is measured in mg/mL.

(a) How rapidly was the BAC increasing after 10 minutes?

(b) How rapidly was it decreasing half an hour later?

Source: Adapted from P. Wilkinson et al., “Pharmacokinetics of Ethanol after Oral Administration in the Fasting State,” Journal of Pharmacokinetics and Biopharmauutics5 (1977): 207- 24.

(a)

To determine

To find: The increase in BAC after 10 minutes.

Explanation

Given:

The resulting data the modeled by the concentration function C(t)=0.0225te0.0467t.

Derivative Rule:

(1) Product Rule: ddx(fg)=fddx(g)+gddx(f)

Result used: Chain Rule

If g is differentiable at t and f is differentiable at g(t), then the composite function F=fg defined by F(t)=f(g(t)) is differentiable at t and F is given by the product

F(t)=f(g(t))g(t) (1)

Calculation:

Obtain the derivative C(t).

C(t)=ddt(C(t))=ddt(0.0225te0.0467t)

Apply the product rule (1),

C(t)=0

(b)

To determine

To find: The decrease in BAC half an hour later.

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