Chapter 7.4, Problem 36E

a.

To determine

To find: Estimate the year the animal died.

Answer to Problem 36E

the year the animal died $=12571\text{ BC}$ .

Explanation of Solution

Given information: A fossilized bone found in central Illinois in the year $2000$

  $\text{C}\text{.E}$ contains $\text{17\%}$ of its original $\text{carbon-14}$ content.

Formula used:

Half-life

  $A=A_0{\left(\frac{1}{2}\right)}^{\frac{t}{h}}$

  $A=$ the amount left after time t

  $A_0=$ initial amount

  $h=$ Half-life

Half-life of $\text{carbon-14}$

  $=5700\text{ Years}$

Calculation:

A fossilized bone found in central Illinois in the year $2000$

  $\text{C}\text{.E}$ contains $\text{17\%}$ of its original $\text{carbon-14}$ content.

Half-life

  $A=A_0{\left(\frac{1}{2}\right)}^{\frac{t}{h}}$

  $A=$ the amount left after time t

  $A_0=$ initial amount

  $h=$ Half-life

Half-life of $\text{carbon-14}$

  $=5700\text{ Years}$

Given

  $\begin{array}{l}A=0.17A_0,h=5700\\ 0.17A_0=A_0{\left(\frac{1}{2}\right)}^{\frac{t}{5700}}\\ {\left(\frac{1}{2}\right)}^{\frac{t}{5700}}=0.17\\ \frac{t}{5700}\ln \frac{1}{2}=\ln 0.17\\ t=\frac{\ln 0.17\times 5700}{\ln \frac{1}{2}}\\ t\approx 14571\end{array}$

The animal died about $14571$ years before the current year $2000$ $\text{C}\text{.E}$

  $2000-14571=-12571\text{ BC}$ .

A negative year means BC

the year the animal died $=12571\text{ BC}$

b.

To determine

To find: Estimate the year the animal died.

Answer to Problem 36E

the year the animal died $=12101\text{ BC}$ .

Explanation of Solution

Given information: A fossilized bone found in central Illinois in the year $2000$

  $\text{C}\text{.E}$ contains $\text{18\%}$ of its original $\text{carbon-14}$ content.

Formula used:

Half-life

  $A=A_0{\left(\frac{1}{2}\right)}^{\frac{t}{h}}$

  $A=$ the amount left after time t

  $A_0=$ initial amount

  $h=$ Half-life

Half-life of $\text{carbon-14}$

  $=5700\text{ Years}$

Calculation:

A fossilized bone found in central Illinois in the year $2000$

  $\text{C}\text{.E}$ contains $\text{18\%}$ of its original $\text{carbon-14}$ content.

Half-life

  $A=A_0{\left(\frac{1}{2}\right)}^{\frac{t}{h}}$

  $A=$ the amount left after time t

  $A_0=$ initial amount

  $h=$ Half-life

Half-life of $\text{carbon-14}$

  $=5700\text{ Years}$

Given

  $\begin{array}{l}A=0.18A_0,h=5700\\ 0.18A_0=A_0{\left(\frac{1}{2}\right)}^{\frac{t}{5700}}\\ {\left(\frac{1}{2}\right)}^{\frac{t}{5700}}=0.18\\ \frac{t}{5700}\ln \frac{1}{2}=\ln 0.18\\ t=\frac{\ln 0.18\times 5700}{\ln \frac{1}{2}}\\ t\approx 14101\end{array}$

The animal died about $14101$ years before the current year $2000$ $\text{C}\text{.E}$

  $2000-14101=-12101\text{ BC}$

A negative year means BC

the year the animal died $=12101\text{ BC}$

c.

To determine

To find: Estimate the year the animal died.

Answer to Problem 36E

the year the animal died $13070\text{ BC}$

Explanation of Solution

Given information: A fossilized bone found in central Illinois in the year $2000$ $\text{C}\text{.E}$ contains $\text{16\%}$ of its original $\text{carbon-14}$ content.

Formula used:

Half-life

  $A=A_0{\left(\frac{1}{2}\right)}^{\frac{t}{h}}$

  $A=$ the amount left after time t

  $A_0=$ initial amount

  $h=$ Half-life

Half-life of $\text{carbon-14}$

  $=5700\text{ Years}$

Calculation:

A fossilized bone found in central Illinois in the year $2000$ $\text{C}\text{.E}$ contains $\text{16\%}$ of its original $\text{carbon-14}$ content.

Half-life

  $A=A_0{\left(\frac{1}{2}\right)}^{\frac{t}{h}}$

  $A=$ the amount left after time t

  $A_0=$ initial amount

  $h=$ Half-life

Half-life of $\text{carbon-14}$

  $=5700\text{ Years}$

Given

  $\begin{array}{l}A=0.16A_0,h=5700\\ 0.16A_0=A_0{\left(\frac{1}{2}\right)}^{\frac{t}{5700}}\\ {\left(\frac{1}{2}\right)}^{\frac{t}{5700}}=0.16\\ \frac{t}{5700}\ln \frac{1}{2}=\ln 0.16\\ t=\frac{\ln 0.16\times 5700}{\ln \frac{1}{2}}\\ t\approx 15070\end{array}$

The animal died about $15070$ years before the current year $2000$ $\text{C}\text{.E}$

  $2000-15070=-13070\text{ BC}$

A negative year means BC

the year the animal died $13070\text{ BC}$