Chapter 1, Problem 1.14P

Write the conversion factor(s) for

1. ${\text{in}}^{\text{2}}{\text{ to m}}^{\text{2}}$ (b) ${\text{km}}^{\text{2}}{\text{ to cm}}^{\text{2}}$

(c) $\text{mi/h to m/s}$ (d) ${\text{ib/ft}}^{\text{3}}{\text{ to g/cm}}^{\text{3}}$

Interpretation Introduction

Interpretation:Conversion factor(s) for ${\text{in}}^{\text{2}}$ to ${\text{m}}^{\text{2}}$ should be determined.

Concept introduction:Conversion factor is ratio that is widely used to describe relation between two different units. In order to convert one unit into another, original unit is multiplied with proper conversion factor to yield new unit. This is done as mentioned below.

  $\text{Final unit}=\left(\cancel{\text{Original unit}}\right)\left(\frac{\text{Final unit}}{\cancel{\text{Original unit}}}\right)$

Explanation of Solution

Both inches and meter are units used for measurement of lengths. Since 1 inch is equivalent to 0.0254 meters, conversion factor for conversion of length from inches to meters is mentioned below.

  $\text{Conversion factor}=\left(\frac{0.0254\text{ m}}{1\text{ in}}\right)$

But the original unit is present in ${\text{in}}^{\text{2}}$ and final unit is required in ${\text{m}}^{\text{2}}$ . For this purpose, above mentioned conversion factor should be squared and required conversion factor for conversion of ${\text{in}}^{\text{2}}$ to ${\text{m}}^{\text{2}}$ is as follows:

  $\begin{array}{c}\text{Conversion factor}={\left(\frac{0.0254\text{ m}}{1\text{ in}}\right)}^2\\ =\left(\frac{0.00064516{\text{ in}}^2}{1{\text{ m}}^2}\right)\end{array}$

Interpretation Introduction

Interpretation:Conversion factor for ${\text{km}}^{\text{2}}$ to ${\text{cm}}^{\text{2}}$ should be determined.

Concept introduction:Conversion factor is ratio that is widely used to describe relation between two different units. In order to convert one unit into another, original unit is multiplied with proper conversion factor to yield new unit. This is done as mentioned below.

  $\text{Final unit}=\left(\cancel{\text{Original unit}}\right)\left(\frac{\text{Final unit}}{\cancel{\text{Original unit}}}\right)$

Explanation of Solution

Both kilometer and centimeter are units used for measurement of lengths. Since 1 kilometer is equivalent to 100000 centimeters, conversion factor for conversion of length from kilometer to centimeter is mentioned below.

  $\text{Conversion factor}=\left(\frac{{10}^5\text{ cm}}{1\text{ km}}\right)$

But the original unit is present in ${\text{km}}^{\text{2}}$ and final unit is required in ${\text{cm}}^{\text{2}}$ . For this purpose, above mentioned conversion factor should be squared and required conversion factor for conversion of ${\text{km}}^{\text{2}}$ to ${\text{cm}}^{\text{2}}$ is as follows:

  $\begin{array}{c}\text{Conversion factor}={\left(\frac{{10}^5\text{ cm}}{1\text{ km}}\right)}^2\\ =\left(\frac{{10}^{10}{\text{ cm}}^2}{1{\text{ km}}^2}\right)\end{array}$

Interpretation Introduction

Interpretation:Conversion factor for $\text{mi/h}$ to $\text{m/s}$ should be determined.

Concept introduction:Conversion factor is ratio that is widely used to describe relation between two different units. In order to convert one unit into another, original unit is multiplied with proper conversion factor to yield new unit. This is done as mentioned below.

  $\text{Final unit}=\left(\cancel{\text{Original unit}}\right)\left(\frac{\text{Final unit}}{\cancel{\text{Original unit}}}\right)$

Explanation of Solution

Both miles and meter are units used for measurement of lengths while hour and second are units for time measurement. Since 1 mile is equivalent to 1609.34 meters and 1 hour is equivalent to 3600 seconds, conversion factor for conversion of $\text{mi/h}$ from $\text{m/s}$ is mentioned below.

  $\text{Conversion factor}=\left(\frac{1609.34\text{ m}}{1\text{ mi}}\right)\left(\frac{1\text{ h}}{3600\text{ s}}\right)$

Interpretation Introduction

Interpretation:Conversion factor for ${\text{lb/ft}}^3$ to ${\text{g/cm}}^{\text{3}}$ should be determined.

Concept introduction:Conversion factor is ratio that is widely used to describe relation between two different units. In order to convert one unit into another, original unit is multiplied with proper conversion factor to yield new unit. This is done as mentioned below.

  $\text{Final unit}=\left(\cancel{\text{Original unit}}\right)\left(\frac{\text{Final unit}}{\cancel{\text{Original unit}}}\right)$

Explanation of Solution

Both pound and mass are units to measure mass while feet and centimeter are used to measure lengths. 1 pound is equivalent to 453.592 grams and 1 feet is equivalent to 30.48 centimeters. Therefore conversion factor for conversion of ${\text{lb/ft}}^3$ to ${\text{g/cm}}^{\text{3}}$ is as follows:

  $\begin{array}{c}\text{Conversion factor}=\left(\frac{453.592\text{ g}}{1\text{ lb}}\right){\left(\frac{1\text{ ft}}{30.48\text{ cm}}\right)}^3\\ =\left(\frac{453.592\text{ g}}{1\text{ lb}}\right)\left(\frac{1{\text{ ft}}^3}{28316.846592{\text{ cm}}^3}\right)\end{array}$