   Chapter 4.6, Problem 17E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Radioactive Decay What percent of a present amount of radioactive radium ( 226 Ra ) will remain after 900 years? (The half-life of 226 Ra is 1599 years.)

To determine

To calculate: The percentage of the radioactive radium R226a with half-life 1599 years that will remain after 900 years.

Explanation

Given Information:

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.

Percentage of the remaining amount =Amount leftInitial amount×100

Calculation:

Consider the provided information that the half-life of R226a is 1599.

Let C be the initial amount of R226a.

As the half-life of R226a is 1599 years, that is, at t=1599 the initial amount reduces to half so,

y=C2

Substitute t=1599 and y=C2 in the equation y=Cekt.

C2=Cek(1599)e1599k=C2C

Take natural log on both the sides,

ln(e1599k)=ln(C2C)1599k(lne)=ln(12)1599k=ln(12)k=ln(12)1599

Hence, the value of k is ln(12)1599

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