Course:MATH102/Question Challenge/2001 December Q03

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Question

Blood alcohol level (BAL), the amount of alcohol in the blood stream (here represented by B(t)), is measured in milligrams of alcohol per 100 milliliters of blood. At the end of a party (time t=0), a drinker is found to have $B(0)=0.08$ (the legal level for driving impairment), and at that time, B(t) satisfies the differential equation

${\frac {dB}{dt}}=-kB$

where k is a positive constant that represents the rate of removal of alcohol from the blood stream by the liver.

(a) If the drinker had waited for three hours before driving, (until t=3), his BAL would have dropped to 0.04. Determine the value of the rate constant k (specify the appropriate units) for this drinker.

(b) According to this model, how much longer would it take for the BAL to drop to 0.01?

Hints

Hint
What kind of function $B(t)$ satisfies the differential equation given here?

Hint 2

If the function is of the form

B(t)=Ce^{kt}

, what information can you use to determine the value of the constant C?

Hint 3

Plug in the information provided at

t=3

into the function

B(t)

and solve for k. You will need to

apply $\ln$ to both sides of the equation to solve for k.

Full Solutions

Solution

a) 0.04 = 0.08e^(-k*3)

   ln(0.5)/-3 = k 
   k = 0.231.

b) 0.01 = 0.08e^(-0.231t)

   ln(0.01/0.08)/-0.231 = t 
   t = 9 hours

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