Chapter 1, Problem 90AP

The surface area and average depth of the Pacific Ocean are $\text{1}{\text{.8 × 10}}^{\text{8}}{\text{km}}^{\text{2}}$ and $\text{3}{\text{.9 × 10}}^{\text{2}}\text{m}$. respectively. Calculate the volume of water in the ocean in liters.

Interpretation Introduction

Interpretation:

The volume of water in the Pacific Ocean in liters is to be calculated.

Concept Introduction:

Dimensional analysis is used to set up and solve a unit conversion problem using conversion factors.

The conversion factor is a fraction obtained from a relationship between the units. It is written as a ratio and can be inverted to give two conversion factors for every relationship.

The appropriate conversion factor for any equality is selected in such a way so that it results in the proper unit cancellation.

The relation between cubic meters and liters is expressed as:

$1{\text{ m}}^3=1000\text{ L}$

For conversion of cubic meters, m³ to liters, the conversion factor is $\frac{1000\text{ L}}{1{\text{ m}}^3}$

The relation between kilometers and meters is expressed as:

$1\text{ km}=1000\text{ m}$

For conversion of km to meters, the conversion factor is $\frac{1000\text{ m}}{1\text{ km}}$

The volume is related to surface area and depth as:

$\text{v}=(s)(d)$

Wherevis the volume in liters, $s$ is the surface area, and $d$ is the depth.

Answer to Problem 90AP

Solution: $7.0\times {10}^{20}\text{ L}$

Explanation of Solution

Given information:

The surface area of the oceanis $1.8\times {10}^8{\text{ km}}^2$

Depth of the ocean is $3.9\times {10}^3\text{ m}$

The conversion factors are derived from the equalities $1{\text{ m}}^3=1000\text{ L}$ and $1\text{ km}=1000\text{ m}$.

The appropriate conversion factors for the equalities are $\frac{1000\text{ L}}{1{\text{ m}}^3}$ and $\frac{1000\text{ m}}{1\text{ km}}$.

Convert the surface area from km² to m² as:

$\begin{array}{l}1\text{ km}=1000\text{ m}\\ \Rightarrow 1.8\times {10}^8{\text{ km}}^2=1.8\times {10}^8{\text{ km}}^2\times {\left(\frac{1000\text{ m}}{1\text{ km}}\right)}^2\\ =1.8\times {10}^8\cancel{{\text{km}}^2}\times \frac{{10}^6{\text{ m}}^2}{1\cancel{{\text{km}}^2}}\\ =1.8\times {10}^{14}{\text{m}}^2\end{array}$

The volume of the ocean (in liters) can be evaluated as follows:

$v=(s)(d)$

Substitute $1.8\times {10}^{14}{\text{ m}}^2$ for $s$ and $3.9\times {10}^3\text{ m}$ for $d$.

$\begin{array}{c}v=\left(1.8\times {10}^{14}{\text{ km}}^2\right)\left(3.9\times {10}^3\text{ m}\right)\\ =7.0\times {10}^{17}{\text{m}}^{\text{3}}\end{array}$

Convert the volume from cubic meters to liters as: $\begin{array}{c}7.0\times {10}^{17}{\text{m}}^{\text{3}}=7.0\times {10}^{17}\cancel{{\text{m}}^3}\times \frac{1000\text{}\text{L}}{\text{1 }\cancel{{\text{m}}^3}}\\ =7.0\times {10}^{20}\text{ L}\end{array}$

Conclusion

Hence, the volume of the Pacific Ocean is $7.0\times {10}^{20}\text{ L}$.