Chapter 1, Problem 66QP

Interpretation Introduction

Interpretation:

The conversions are to be carried out by dimensional analysis for given units of measurement.

Concept introduction:

Dimensional analysis is a way to convert the units of measurement. In order to convert one unit to another, one needs to know the relationship between those units. These relationships are called conversion factors. Dimensional analysis is used to set up and solve a unit conversion problem using conversion factors.

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.

$\text{1 yd = 36 in}$, to convert yd to in, the conversion factor is $\frac{36\text{ in}}{1\text{ yd}}$.

$\text{1 in = 2}\text{.54 cm}$, to convert in to cm, the conversion factor is $\frac{2.54\text{ cm}}{1\text{ in}}$.

$\text{1 in = 2}\text{.54 cm}$, to convert cm to in, the conversion factor is $\frac{1\text{ in}}{2.54\text{ cm}}$.

$\text{1 ft}=\text{12 in}$, to convert in to ft, the conversion factor is $\frac{1\text{ ft}}{12\text{ in}}$.

$1\text{ hr}=60\text{ min}$, to convert hours to minutes, the conversion factor is $\frac{60\text{ min}}{1\text{ hr}}$.

$1\text{ min}=60\text{ s}$, to convert minute to second, the conversion factor is $\frac{60\text{s}}{1\min }$.

$\text{1 mile}=\text{1609 m}$, to convert meter to miles, the conversion factor is $\frac{\text{1 mile}}{\text{1609 m}}$.

Answer to Problem 66QP

Solution:

a)

$\text{2}\text{.96}\times {\text{10}}^3\text{cm}$

b)

$\text{9}{\text{.8×10}}^8\text{ ft/s}$

c)

$\text{8}{\text{.35×10}}^{\text{12}}\text{ miles}$

Explanation of Solution

a)32.4 yd to centimeters

The conversion factors are as follows:

$\text{1 yd = 36 in}$

$\text{1 in = 2}\text{.54 cm}$

Now, set up the unit conversion calculation using the appropriate conversion factors.

The conversion is as follows:

$\left(32.4\cancel{\text{yd}}\times \frac{36\cancel{\text{in}}}{1\cancel{\text{yd}}}\times \frac{2.54\text{ cm}}{1\cancel{\text{in}}}\right)=\text{2}\text{.96}\times {\text{10}}^3\text{ cm}$.

The units in the conversion factor must cancel to give the correct unit for the answer.

Hence,

$\text{32}\text{.4 yd}=\text{ 2}\text{.96}\times {\text{10}}^3\text{ cm}$.

b) $\text{3}\text{.0}\times {\text{10}}^{10}\text{ cm/s to ft/s}$

The conversion factors are as follows:

$\text{1 in}=\text{2}\text{.54 cm}$.

$\text{1 ft}=\text{12 in}$.

Now, set up the unit conversion calculation using the appropriate conversion factors.

The conversion is as follows:

$\left(3.0\times {10}^{10}\cancel{\text{cm}}\text{/s × }\frac{1\cancel{\text{in}}}{2.54\cancel{\text{cm}}}\text{ × }\frac{1\text{ ft}}{12\cancel{\text{in}}}\right)=\text{9}{\text{.8×10}}^8\text{ ft/s}$

The units in the conversion factor must cancel to give the correct unit for the answer.

Hence,

$\text{3}\text{.0}\times {\text{10}}^{10}\text{ cm/s}=\text{9}{\text{.8×10}}^8\text{ ft/s}$.

c) $1.42\text{ light years to miles}$

First, convert the speed of light in m/s to mph.

The conversion is as follows:

$\left(\text{3}{\text{.00×10}}^{\text{8}}\cancel{\text{m}}\text{/}\cancel{\text{s}}\text{ × }\frac{\text{1 mi}}{\text{1609 }\cancel{\text{m}}}\text{ × }\frac{\text{60 }\cancel{\text{s}}}{\text{1 }\cancel{\text{min}}}\text{ × }\frac{\text{60 }\cancel{\text{min}}}{\text{1 h}}\right)=\text{6}{\text{.7122×10}}^{\text{8}}\text{ mph}$

The number of hours in a year is calculated as follows:

$\left(365\text{ days}\right)\left(24\text{ h }\right)=8760\text{ h}$

The miles in one light year is calculated as follows:

$\begin{array}{c}\text{1 light year}=\left(\text{6}{\text{.7122×10}}^{\text{8}}\text{ mph}\right)\left(\text{8760 h}\right)\\ =\text{5}{\text{.8799×10}}^{\text{12}}\text{ mi}\end{array}$

The final conversion is as follows:

$\left(\text{1}\text{.42}\cancel{\text{light year}}\right)\left(\frac{\text{5}{\text{.8799×10}}^{\text{12}}\text{ miles}}{\text{1}\cancel{\text{light year}}}\right)=\text{8}{\text{.35×10}}^{\text{12}}\text{ miles}$

Hence,

$1.42\text{ light years}=8.35\times {10}^{12}\text{ miles}$.