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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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Carbon Dating In Exercises 79–82, you are given the ratio of carbon isotopes to carbon atoms in a fossil. Use the information to estimate the age of the fossil. In living organic material, the ratio of radioactive carbon isotopes to the total number of carbon atoms is about 1 to 1012. (See Example 3 in Section 4.1.) When organic material dies, its radioactive carbon isotopes begin to decay, with a half-life of about 5715 years. So, the ratio R of carbon isotopes to carbon-14 atoms is modeled by R = 10 12 ( 1 2 ) t / 5715 , where t is the time (in years) and t = 0 represents the time when the organic material died.

R = 0.32 × 10 12

To determine

To calculate: The age of the fossil when the ratio of carbon isotopes to carbon atoms in a fossil is R=0.32×1012 and the formula to calculate the ratio R is given by R=1012(12)t5715, where t is the time in years and t=0 is the time when the organic material died.

Explanation

Given Information:

The ratio of carbon isotopes to carbon atoms in a fossil is R=0.32×1012 and the formula to calculate the ratio R is given by R=1012(12)t5715, where t is the time in years and t=0 is the time when the organic material died.

Formula used:

The property of logarithms:

lnxn=nlnx

Where x and y are real numbers greater than 0.

Calculation:

Consider the provide information that the ratio of carbon isotopes to carbon atoms in a fossil is R=0.32×1012 and the formula to calculate the ratio R is given by R=1012(12)t5715, where t is the time in years and t=0 is the time when the organic material died.

Consider the equation, R=1012(12)t5715

Here, R=0.32×1012

Calculate the age of fossil by substituting the value of R in the equation as,

0.32×1012=1012(12)t5715

Divide both sides by 1012 as,

0.32×10121012=1012(12)t571510120

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