Chapter 26, Problem 2P

If you represent Earth’s history by a line that is 1 m long, how long a segment would represent the 400 million years since life first moved onto the land? How long a segment would represent the 4-millionyear history of humanoid life?

To determine

The length of the segment that would represent $400\text{milliion}$ years if the History of the Earth is represented by a line that is $1\text{m}$ long ,also find the length of the line that would represent the $4\text{milliion}$ years of humanoid life.

Answer to Problem 2P

The length of the segment for the history of the Earth is calculated as $8.8\text{cm}$ and the length of the segment for the history of the human life is $0.88\text{mm}$.

Explanation of Solution

The expression to calculate the length of the line segment is,

    $\frac{L}{4.6\times {10}^9\text{years}}=\frac{l}{T}$        (I)

Here, $l$ is the length of the segment for Earth’s history, $L$ is the length of the Earth’s history and $T$ is the total time of the history of human life

The expression to calculate the length of the line segment is,

    $\frac{L}{4.6\times {10}^9\text{years}}=\frac{l_h}{T}$        (II)

Here, $l_h$ is the length for the time period of history of humanity.

Conclusion:

Substitute $1.0\text{m}$ for $L$ and $400\text{million years}$ for $T$ in equation (I) to find $l$

    $\begin{array}{c}\frac{1.0\text{m}}{4.6\times {10}^9\text{years}}=\frac{l}{400\text{million years}}\\ \frac{1.0\text{m}}{4.6\times {10}^9\text{years}}=\frac{l}{\left(400\text{million years}\times \frac{{10}^6\text{years}}{1\text{million years}}\right)}\\ \frac{1.0\text{m}}{4.6\times {10}^9\text{years}}=\frac{l}{\left(400\times {10}^6\text{years}\right)}\\ l=\frac{\left(1.0\text{m}\right)\left(400\times {10}^6\text{years}\right)}{4.6\times {10}^9\text{years}}\end{array}$

Solve further as,

    $\begin{array}{c}l=0.88\times {10}^{-1}\times \frac{{10}^2\text{cm}}{1\text{m}}\\ =8.8\text{cm}\end{array}$

Substitute $1.0\text{m}$ for $L$  and $4\text{million years}$ for $T$  in equation (II) to find $l_h$ .

    $\begin{array}{c}\frac{1.0\text{m}}{4.6\times {10}^9\text{years}}=\frac{l_h}{400\text{million years}}\\ \frac{1.0\text{m}}{4.6\times {10}^9\text{years}}=\frac{l_h}{\left(4\text{million years}\times \frac{{10}^6\text{years}}{1\text{million years}}\right)}\\ \frac{1.0\text{m}}{4.6\times {10}^9\text{years}}=\frac{l_h}{\left(4\times {10}^6\text{years}\right)}\\ l_h=\frac{\left(1.0\text{m}\right)\left(4\times {10}^6\text{years}\right)}{4.6\times {10}^9\text{years}}\end{array}$

Solve further as,

    $\begin{array}{c}l=0.88\times {10}^{-3}\text{m}\times \frac{1\text{mm}}{{10}^{-3}\text{m}}\\ =0.88\text{mm}\end{array}$

Therefore, the length of the segment for the history of the Earth is calculated as $8.8\text{cm}$ and the length of the segment for the history of the human life is $0.88\text{mm}$.