EBK PRECALCULUS W/LIMITS
EBK PRECALCULUS W/LIMITS
4th Edition
ISBN: 9781337516853
Author: Larson
Publisher: CENGAGE CO
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Differential Equations
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Chapter 3.5, Problem 22E
To determine

To find : the amount of radioactive isotope after 1000 years.

Expert Solution & Answer
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Answer to Problem 22E

The amount of radioactive isotope after 1000 years is   A≈5.7592 grams    _

Explanation of Solution

Given information : The radioactive isotope details given in the table

    IsotopeHalf-life (years)Initial QuantityAmount After 1000 Years
    14C57156.5 g?

Concept Involved:

Solving for a variable means getting the variable alone in one side of the equation by undoing whatever operation is done to it.

Formula Used:

For Radioactive Decay, the amount A after t years, with initial amount A0 and decay rater (in decimal form) is given by the formula: A = A0ert

Logarithmic property: lneX=X

Calculation:

    DescriptionSteps
    Substitute the given information in the formula A = A0ert { A0=6.5 } A=6.5ert
    To find the decay rate, we need to replace 5 in place of A& t=5715 {Because half-life is 5715years}3.25=6.5e5715r
    Use symmetric property of equality which states that if a = b then b = a to rewrite the equation6.5e5715r=3.25
    Divide by 6.5 on both sides6.5e5715r6.5=3.256.5
    Simplify fraction on both sides of the equatione5715r=0.5
    Take natural logarithm on both sideslne5715r=ln(0.5)

Calculation (Continued):

    DescriptionSteps
    Apply the logarithmic rule lneX=X in left side of the equation 5715r=ln0.5
    Divide by 5715 on both sides 5715r5715=ln0.55715
    Simplify fraction on both sides of the equationr=−0.0001213
    Rewrite the decimal as percentager=−0.0001212⋅100%    =−0.01212%
    Replace 6.5 for A0 &-0.0001212 for r in the equation A = A0ertA =6.5e−0.000121t
    To find the amount after 1000 years substitute 1000 for t in the equationA=6.5e−0.000121( 1000)      =6.5e−0.121      ≈5.7592

Conclusion:

    IsotopeHalf-life (years)Initial QuantityAmount After 1000 Years
    14C57156.5 g5.7592 grams
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Chapter 3.5, Problem 21E
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Chapter 3.5, Problem 23E

Chapter 3 Solutions

EBK PRECALCULUS W/LIMITS

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