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A First Course In Differential Equations With Modelling Applications

Dennis G. Zill

$Math Studyset Vaia Explanations$ Math

11th Edition · 400 pages

ISBN: 9781305965720

Chapter 3: Q12E (page 92)

The Shroud of Turin, which shows the negative image of thebody of a man who appears to have been crucified, is believedby many to be the burial shroud of Jesus of Nazareth. SeeFigure 3.1.12. In 1988 the Vatican granted permission tohave the shroud carbon-dated. Three independent scientificlaboratories analyzed the cloth and concluded that the shroud wasapproximately 660 years old, an age consistent with its historicalappearance. Using this age, determine what percentage of theoriginal amount of C-14 remained in the cloth as of 1988.

Short Answer

Expert verified

The original amount of C-14 is 92.3%.

Step by step solution

01

Define carbon dating.

The characteristics of radiocarbon, a radioactive isotope of carbon, are used in radiocarbon dating to determine the age of an object including organic material.

02

Solve for the original amount of C-14.

To obtain the original amount of C-14 by the equation in page 87.

$A (t) = A_{0} e^{- \frac{\ln 2}{5730} t}$

A cloth was analyzed by three independent scientist’s laboratories, then the age founded is $660$ years. As the age of the cloth is $t = 660$ years, then the equation becomes,

$\begin{array}{rcl} A & = & A_{0} e^{- \frac{\ln 2}{π 730} \times 660} \\ = & A_{0} e^{- 0.07984} \\ = & 0.923 A_{0} \\ A & = & 92.3 percent \end{array}$

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Most popular questions from this chapter

Two chemicalsandare combined to form a chemical. The rate, or velocity, of the reaction, is proportional to the product of the instantaneous amounts ofand B not converted to chemical C. Initially, there aregrams of A and 50 grams of B, and for each gram of B,grams of A is used. It is observed that 10 grams of C are formed in 5 minutes. How much is formed in minutes? What is the limiting amount of C after a long time? How much of chemicals A and B remain after a long time?

When forgetfulness is taken into account, the rate of memorization of a subject is given by

$\frac{d A}{d t} = k_{1} (M - A) - k_{2} A$

Where, $k_{1} > 0, k_{2} > 0, A (t)$ is the amount memorized in time t, M is the total amount to be memorized, and $M - A$ is the amount remaining to be memorized.

(a) Since the DE is autonomous, use the phase portrait concept of Section 2.1 to find the limiting value of $A (t)$ as $t \to \infty$ . Interpret the result.

(b) Solve the DE subject to $A (0) = 0$ . Sketch the graph of $A (t)$ and verify your prediction in part (a).

(a) Consider the initial-value problem $d A / d t = k A, A (0) = A_{0}$ as the model for the decay of a radioactive substance.Show that, in general, the half-life Tof the substance is $T = - (l n 2) / k$ .

(b) Show that the solution of the initial-value problem in part(a) can be written localid="1663839717694" $A (t) = A_{0} 2^{- t / T}$

(c) If a radioactive substance has the half-life Tgiven in part (a),how long will it take an initial amount $A_{0}$ of the substance todecay to how long will it take an initial amount $\frac{1}{8} A_{0}$ ?

Population Model In one model of the changing population

P(t) of a community, it is assumed that

$\frac{d P}{d t} = \frac{d B}{d t} = \frac{d D}{d t}$

where $\frac{dB}{dt}$ and $\frac{dD}{dt}$ are the birth and death rates, respectively.

(a) Solve for P(t) if $\frac{d B}{d t} = k_{1} P$ and $\frac{dD}{dt} = k_{2} P$

(b) Analyze the cases $k_{1} > k_{2}, k_{1} = k_{2} a n d k_{1} < k_{2}$

A dead body was found within a closed room of a house where the temperature was a constant $70^{o} F$ . At the time of discovery the core temperature of the body was determined to be $85^{o} F$ . One hour later a second measurement showed that the core temperature of the body was $80^{o} F$ . Assume that the time of death corresponds to $t = 0$ and that the core temperature at that time was $98 . 6^{o} F$ .Determine how many hours elapsed before the body was found.[Hint: Let $t_{1} > 0$ denote the time that the body was discovered.]

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