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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Radioactive Decay In Exercises 29-36, complete the table for the radioactive isotope.

Isotope Half-life (in years) Initial Quantity Amount After 1000 Years Amount After 10,000 Years

14 C

5715

_____ _____ 3 g

To determine

To calculate: The initial amount of the radioactive isotope 14C and its total amount after 1000 years.

Explanation

Given:

Half-life of the radioactive isotope 14C as 5715 and its amount after 10,000 years as 3g

Formula used:

We can use growth and decay model y=Cekt.

Logarithmic property:

ln(ex)=x

Also,

eln(x)=x

Calculation:

Let us consider the given expression:

y=Cekt …… (1)

Let the initial amount be C.

t=10000 for the amount after 10,000 years and also given that y=3.

By putting t=10000 and y=3 in equation (1), we get:

y=Cekt3=Cek×10000C=3e10000k

For half-life, t=5715 and y=12C.

By putting t=5715 and y=12C in equation (1), we get:

y=Cekt12C=Cek×571512=e5715kln(12)=ln(e5715k)

By simplifying further,

5715k=ln(12)5715k=0

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