Answered: Amount Amount Half-life Initial After…

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Amount Amount Half-life Initial After After Isotope (in years) Quantity 1000 Years 10,000 Years 32. 14С 5715 3 8

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.CT: Chapter Test

Problem 14CT

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Complete the table for the radioactive isotope.

Amount
Amount
Half-life Initial
After
After
Isotope (in years) Quantity 1000 Years 10,000 Years
32. 14С
5715
3 g

Expert Solution

Step 1

Recall the following.

The exponential decay equation is $A (t) = A_{0} e^{- r t}$ .

The half-life of the given isotope is 5715 years.

Obtain the value of r as follows.

$\frac{A_{0}}{2} = A_{0} e^{- r (5715)} e^{- r (5715)} = \frac{1}{2} - 5715 r = \ln (\frac{1}{2}) r = - \frac{\ln (\frac{1}{2})}{5715}$

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