Chapter 2, Problem 31E

Interpretation Introduction

Interpretation:

Each number in given unit is to be determined.

Concept Introduction:

Dimensional analysis is useful in converting one unit to another.

To convert a quantity $a{u}_1$ from one unit to another, that is, $u_1\to u_2$, first write a conversion factor $\frac{u_2}{u_1}$, then multiply the conversion factor with the given quantity $a{u}_1$, cancelling units to get the quantity in the desired unit as follows:

$a\cancel{u_1}\times \frac{u_2}{\cancel{u_1}}=a{u}_2$

Answer to Problem 31E

Solution:

(a)

${10}^6{\text{m}}^2$

(b)

$2.82\times {10}^4{\text{ cm}}^3$

(c)

${\text{9 ft}}^2$

Explanation of Solution

a) Square meters (${\text{m}}^2$) in 1 square kilometer (${\text{km}}^2$).

To convert ${\text{km}}^2$ to ${\text{m}}^2$, first write the correct conversion factor by using the following relationship:

$\begin{array}{c}1\text{ km}=1000\text{ m}\\ {\left(\text{1km}\right)}^2={\left(\text{1000 m}\right)}^2\\ 1{\text{ km}}^2=1000000{\text{ m}}^2\end{array}$

Thus, correct conversion factor will be $\left(\frac{{\text{1000000 m}}^2}{1{\text{ km}}^2}\right)$.

Multiply the conversion factor with the given quantity, cancelling units to get the desired quantity.

Hence,

$1\cancel{{\text{km}}^2}\times \frac{1000000m^2}{1\cancel{{\text{km}}^2}}={10}^6{\text{ m}}^2$

b) Cubic centimeters (${\text{cm}}^3$) in 1 cubic foot (${\text{ft}}^3$).

To convert ${\text{ft}}^3$ to ${\text{cm}}^3$, first write the correct conversion factor by using the following relationship:

$\begin{array}{c}1\text{ ft}=12\text{ in}\text{.}\\ {\left(\text{1ft}\right)}^3={\left(\text{12 in}\right)}^3\\ 1{\text{ ft}}^3=1728{\text{ in}}^3\end{array}$

And

$\begin{array}{c}1\text{ in}\text{.}=2.54\text{ cm}\\ {\left(\text{1in}\text{.}\right)}^3={\left(\text{2}\text{.54 cm}\right)}^3\\ 1{\text{ in}}^3=16.387{\text{ cm}}^3\end{array}$

Thus, correct conversion factors will be $\left(\frac{1726\text{ in}{\text{.}}^3}{{\text{1 ft}}^3}\right)\text{and}\left(\frac{\text{16}{\text{.387 cm}}^3}{{\text{1 in}}^3}\right)$.

Multiply the conversion factors with the given quantity, cancelling units to get the desired quantity.

Hence,

$1\cancel{{\text{ ft}}^3}\times \frac{1726\cancel{{\text{in}}^3}}{1\cancel{{\text{ft}}^3}}\times \frac{\text{16}{\text{.387 cm}}^3}{1\cancel{{\text{in}}^3}}=2.82\times {10}^4{\text{ cm}}^3$

c) Square feet (${\text{ft}}^2$) in 1 square yard (${\text{yd}}^2$).

To convert ${\text{yd}}^2$ to ${\text{ft}}^2$, first write the correct conversion factor by using the following relationship:

$\begin{array}{c}1\text{ yd}=3\text{ ft}\\ {\left(\text{1yd}\right)}^2={\left(\text{3 ft}\right)}^2\\ 1{\text{ yd}}^2=9{\text{ ft}}^2\end{array}$

Thus, correct conversion factor will be $\left(\frac{{\text{9 ft}}^2}{1{\text{ yd}}^2}\right)$.

Multiply the conversion factor with the given quantity, cancelling units to get the desired quantity.

Hence,

$1\cancel{{\text{yd}}^2}\times \frac{9f{t}^2}{1\cancel{{\text{yd}}^2}}=9{\text{ ft}}^2$