Chapter 4.6, Problem 18E

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

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

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

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Radioactive Decay Find the half-life of a radioactive material for which 99.57% of the initial amount remains after 1 year.

To determine

To calculate: The half-life of a radioactive material for which 99.57% of initial amount remains after 1 year.

Explanation

Given Information:

The provided information is that 99.57% of initial amount remains after 1 year.

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 99.57% of initial amount remains after 1 year.

Let C be the initial amount of the radioactive material and h be the half-life of the radioactive material.

As 99.57% of initial amount remains after 1 year, so at t=1 the remaining amount is,

y=99.57% of C=99.57100×C=0.9957×C=0.9957C

Substitute t=1 and y=0.9957C in the equation y=Cekt.

0.9957C=Cek10.9957C=Cek

Also, the half-life of the radioactive material is h, that is, at t=h the initial amount reduces to half so,

y=C2

Substitute t=h and y=C2 in the equation y=Cekt

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