Chapter 7.5, Problem 39P

To estimate the age of an organism, archaeologists measure the amount of carbon- 14 left in its remains. The approximate amount of carbon- 14 remaining after 5000 years can be found using the formula $A=A_0{(2.7)}^{-\frac{3}{5}}$

where $A_0$ is the initial amount of carbon- 14 in the sample that is tested. How much carbon- 14 is left in a sample that is 5000 years old and originally contained $5.0\times {10}^{-3}$ grams of carbon-14? Write your answer in scientific notation.

To determine

To calculate: the approximate amount of carbon -14 remaining after 5000 years in scientific notation and originally contained $5.0\times {10}^{-3}$ grams of carbon -14.

Answer to Problem 39P

The approximated amount of carbon -14 after 5000 years = $1.738408135\times {10}^{-4}$ grams

Explanation of Solution

Given:

He have given that initial amount of carbon -14 in a sample that is 5000 years old

  =$5.0\times {10}^{-3}$ grams

Formula used:

Also he have given the formula used for this problem is

  $A=A_0.{(2.7)}^{-3/5}$

Where A₀ is the initial amount of carbon -14 in a sample that is 5000 years old .

AndA is the approximated value of carbon -14 that we have to calculate.

Scientific Notation rules or definition

The proper format for scientific notation is $a\times {10}^b$ where a is a number or decimal number such that the absolute value of a is greater than or equal to one and less than ten or,

  $1 \le \left|a\right| < 10$ And b is the power of 10 required so that the scientific notation is mathematically equivalent to the original number.

Calculation :

Hence now we put the initial amount of carbon -14 in the above formula

We getA= $5.0\times {10}^{-3}.{\left(2.7\right)}^{-3/5}$

  =$5.0\times {10}^{-3}.{\left(27/10\right)}^{-3/5}$

  =$5.0\times {10}^{-3}{.10}^{-3/5}{\left(27\right)}^{-3/5}$

  =$5.0\times {10}^{-3-3/5}{\left(3^3\right)}^{-3/5}$

  =$5.0\times {10}^{-18/5}{\left(3\right)}^{-9/5}$

  $\begin{array}{l}=5.0\times {\left({10}^2\right)}^{-9/5}{\left(3\right)}^{-9/5}\hfill \\ =5.0\times {\left({10}^2.3\right)}^{-9/5}\hfill \\ =5.0\times {\left(300\right)}^{-9/5}\hfill \\ =5.0\times 3.47681627\times {10}^{-5}\hfill \\ =17.38408135\times {10}^{-5}\hfill \\ =1.738408135\times {10}^{-4 }\hfill \end{array}$

which is the required scientific notation.

Conclusion:

Hence approximated value of carbon -14 after 5000 years= $1.738408135\times {10}^{-4}$