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Chapter 6: Problem 17

Medicine The rate of change with respect to time of the quantity \(q\) of pain reliever in a person's body \(t\) hours after the individual takes the medication is proportional to the quantity of medication remaining. Assume that 2 hours after a person takes 200 milligrams of a pain reliever, onehalf of the original dose remains. a. Write a differential equation for the rate of change of the quantity of pain reliever in the body. b. Find a particular solution for this differential equation. c. How much pain reliever will remain after 4 hours; after 8 hours?

Short Answer

Expert verified
After 4 hours, 50 mg remains; after 8 hours, 12.5 mg remains.

Step by step solution

01

Formulate the Differential Equation

The problem states that the rate of change of the quantity of pain reliever, \( q(t) \), with respect to time, \( t \), is proportional to the quantity remaining. Mathematically, this is expressed as \( \frac{dq}{dt} = -kq \), where \( k > 0 \) is the constant of proportionality.
02

Solve the Differential Equation

This is a separable differential equation. We rewrite it as \( \frac{1}{q} \frac{dq}{dt} = -k \), integrate both sides: \(\int \frac{1}{q} \, dq = -k \int dt \), resulting in \( \ln |q| = -kt + C \). Solving for \( q \), we have \( q(t) = Ce^{-kt} \), where \( C \) is the integration constant.
03

Determine the Constant C Using Initial Conditions

We know that initially, \( q(0) = 200 \). Thus, \( 200 = C \cdot e^{0} \) implies \( C = 200 \). So the solution becomes \( q(t) = 200e^{-kt} \).
04

Use Given Condition to Find the Rate Constant k

It's given that after 2 hours, half of the original dose remains, so \( q(2) = 100 \). Substituting, \( 100 = 200e^{-2k} \), solving for \( k \), we get \( e^{-2k} = 0.5 \). Taking the natural logarithm, \( -2k = \ln(0.5) \), resulting in \( k = -\frac{1}{2} \ln(0.5) \).
05

Evaluate for 4 and 8 Hours

Substitute \( k = -\frac{1}{2} \ln(0.5) \) into the model \( q(t) = 200e^{-kt} \). For \( t = 4 \) hours, \( q(4) = 200 \times e^{-2k} \), which simplifies to 50 mg. For \( t = 8 \) hours, \( q(8) = 200 \times e^{-4k} \), which simplifies to 12.5 mg.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Decay
Exponential decay is a mathematical process where quantities decrease over time at a rate proportional to their current value. This is seen in the way the pain reliever dissolves in the body. As described in the medicine exercise, the reliever's concentration diminishes in a similar fashion. The rate of change is captured by the equation \( \frac{dq}{dt} = -kq \), reflecting how each moment, the remaining quantity decreases in proportion to its current amount.
Exponential decay occurs often in nature and various scientific fields:
  • Radioactive decay, where unstable atoms lose particles.
  • Cooling of a hot object, where heat escapes to the cooler surroundings.
  • The discharge of a capacitor, showing how voltage decreases over time.
By understanding exponential decay, we learn how substances like medicine release their effect over time, and why careful timing of dosages is crucial.
Pharmacokinetics
Pharmacokinetics is the study of how drugs move through the body over time. It covers how substances enter, circulate, and leave the system. This exercise mimics a real-world pharmacokinetic scenario with a pain reliever's concentration decreasing exponentially.
Key processes involved in pharmacokinetics include:
  • Absorption: How the drug enters the bloodstream.
  • Distribution: How it spreads through body tissues.
  • Metabolism: The chemical alteration of the drug.
  • Excretion: The elimination of the drug from the body.
In the exercise, the focus is on the rate at which the medication depletes, showcasing the elimination aspect. Understanding pharmacokinetics can inform medical professionals about dose adjustments necessary for safe and effective treatment.
Separable Differential Equations
A separable differential equation is a common type of differential equation that can be split into two functions, each in one variable only. Solving involves integrating these two separate parts.
In the case presented, the differential equation \( \frac{dq}{dt} = -kq \) is separable. The steps to solve it included separating the variables, integrating, and then rearranging to find the solution. Steps are as follows:
  • Rearrange: \( \frac{1}{q} \frac{dq}{dt} = -k \)
  • Integrate both sides: \( \int \frac{1}{q} \, dq = -k \int dt \)
  • Solve: Resulting in \( \ln|q| = -kt + C \)
  • Exponentiate to remove \( \ln \): \( q(t) = Ce^{-kt} \)
Separable differential equations like this one allow for simple solutions in exponential models, exemplifying how basic calculus techniques apply to real-world problems.

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Most popular questions from this chapter

Suppose that \(g\) with input \(t\) is an exponential density function with \(k=2\). a. Find \(G,\) the corresponding cumulative distribution function. b. Use both \(g\) and \(G\) to calculate the probability that \(t \leq 0.35\) c. Use \(G\) to calculate the probability that \(t>0.86\).

Write an equation or differential equation for the given information. The Verhulst population model assumes that a population \(P\) in a country will be increasing with respect to time \(t\) at a rate that is jointly proportional to the existing population and to the remaining amount of the carrying capacity \(C\) of that country.

Postage Stamps In 1880,37 countries issued postage stamps. The rate of change (with respect to time) of the number of countries issuing postage stamps between 1836 and 1880 was jointly proportional to the number of countries that had already issued postage stamps and to the number of countries that had not yet issued postage stamps. The constant of proportionality was approximately 0.0049. By 1855,16 countries had issued postage stamps. (Source: "The Curve of Cultural Diffusion," American Sociological Review, August \(1936,\) pp. \(547-556)\) a. Write a differential equation describing the rate of change in the number of countries issuing postage stamps with respect to the number of years since 1800 b. Write a general solution for the differential equation. c. Write the particular solution for the differential equation. d. Estimate the number of countries that were issuing postage stamps in 1840 and in \(1860 .\)

Write a differential equation expressing the information given and, when possible, find a general solution for the differential equation. The rate of change with respect to time \(t\) of the demand \(D\) for a product is decreasing in proportion to the demand at time \(t\).

Plow Patents The number of patents issued for plow sulkies between 1865 and 1925 was increasing with respect to time at a rate jointly proportional to the number of patents already obtained and to the difference between the number of patents already obtained and the carrying capacity of the system. The carrying capacity was approximately 2700 patents, and the constant of proportionality was about \(7.52 \cdot 10^{-5} .\) By 1883,980 patents had been obtained. (Source: Hamblin, Jacobsen, and Miller, \(A\) Mathematical Theory of Social Change, New York: Wiley, 1973 ) a. Write a differential equation describing the rate of change in the number of patents with respect to the number of years since 1865 b. Write a general solution for the differential equation. c. Write the particular solution for the differential equation. d. Estimate the number of patents obtained by 1900 .
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