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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Modeling Radioactive Decay In Exercises 121-126, complete the table for each radioactive isotope.

Amount Amount
after after
Half-life Initial 1000 10,000
Isotope (in years) quantity years years

14 C

5715 5 grams

To determine

To calculate: The amount of C14 left after 1000 years and 10,000 years, if the half-life of C14 is 5715 years and the initial quantity is 5 grams.

Explanation

Given Information:

The provided half-life of C14 is 5715 years and the initial quantity is 5 grams.

Formula used:

Formula for ln is:

ln(x)n=nln(x)

And lne=1

Calculation:

Consider the provided information that the half-life of C14 is 5715 years and the initial quantity is 5 grams.

Observe that here at t=0, the initial amount y=5.

Hence, the value of C is 5.

Substitute C=5 in the equation y=Cekt.

y=5ekt

Also, the half-life of C14 is 5715 years, that is, at t=5715 the initial amount reduces to half so,

y=52=2.5

Substitute t=5715 and y=2.5 in the equation y=5ekt.

2.5=5ek(5715)e5715k=2.55

Take natural log on both the sides,

ln(e5715k)=ln2.555715k(lne)=ln(12)5715k=ln(12)k=ln(12)5715

Hence, the value of k is ln(12)5715.

Substitute C=5 and k=ln(12)5715 in the equation y=Cekt

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