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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Medical Science Soon after an injection, the concentration D (in milligrams per milliliter) of a drug in a patient’s bloodstream is 500 milligrams per milliliter. After 6 hours, 50 milligrams per milliliter of the drug remains in the bloodstream.

(a) Find an exponential model for the concentration D after t hours.

(b) What is the concentration of the drug after 4 hours?

(c) Use a graphing utility to graph the function found in part (a).

(a)

To determine

To calculate: The exponential model for the concentration D after t hours of injection in the bloodstream of a patient in milligrams per milliliter if soon after injection it is 500 milligrams per milliliter and 50 milligrams per milliliter is remaining after 6 hours.

Explanation

Given Information:

The concentration D in the bloodstream of a patient after injection it is 500 milligrams per milliliter and 50 milligrams per milliliter is remaining after 6 hours.

Formula used:

Exponential growth and decay:

If the rate of change of a positive quantity y with respect to time is proportional to the amount of quantity present at any time t, that is dydt=ky, then y is given by the equation, y=Cekt, where C is the value of the quantity at time t=0 and k is the constant of proportionality.

If k>0 then there is exponential growth and when k<0 then there is exponential decay.

If the expression is in the form of ex=b, then the property of natural logarithms is given by.

lnb=x

Calculation:

Consider the equation, y=Cekt.

Here, the initial concentration D at t=0 is 500

Substitute in the equation y=Cekt

500=Cek0500=C×1C=500

Evaluate the model after t hours by substituting C=500 in the equation D=Cekt

D=500ekt

Here, the concentration after 6 hours, that is, at t=6 is reduces to, D=50

Substitute t=6 and D=50 in the equation D=500ekt

(b)

To determine

To calculate: The concentration D after 4 hours of injection in the bloodstream of a patient in milligrams per milliliter if soon after injection it is 500 milligrams per milliliter and 50 milligrams per milliliter is remaining after 6 hours.

(c)

To determine

To graph: The function of concentration D after t hours of injection in the bloodstream of a patient in milligrams per milliliter obtained in part (a) if soon after injection it is 500 milligrams per milliliter and 50 milligrams per milliliter is remaining after 6 hours.

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