Chapter 2, Problem 32E

Interpretation Introduction

Interpretation:

Each number for the 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 32E

Solution:

(a)

${10}^{\text{4 }}{\text{cm}}^2$

(b)

$4.6656\times {10}^4{\text{ in}}^3$

(c)

$9.29\times {10}^2{\text{ cm}}^2$

Explanation of Solution

a) Square centimeters (${\text{cm}}^2$) in 1 square meter (${\text{m}}^2$).

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

$\begin{array}{c}1\text{ m}=100\text{ cm}\\ {\left(\text{1m}\right)}^2={\left(\text{100 cm}\right)}^2\\ 1{\text{ m}}^2=10000{\text{ cm}}^2\end{array}$

Thus, correct conversion factor will be $\left(\frac{{\text{10000 cm}}^2}{1{\text{ m}}^2}\right)$.

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

Hence,

$1\cancel{{\text{m}}^2}\times \frac{10000c{m}^2}{1\cancel{{\text{m}}^2}}={10}^4{\text{ cm}}^2$

b) Cubic inches ($\text{in}{\text{.}}^3$) in 1 cubic yard (${\text{yd}}^3$).

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

$\begin{array}{c}1\text{ yd}=3\text{ ft}\\ {\left(\text{1yd}\right)}^3={\left(\text{3 ft}\right)}^3\\ 1{\text{ yd}}^3=27{\text{ ft}}^3\end{array}$

And

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

Thus, correct conversion factors will be $\left(\frac{{\text{27 ft}}^3}{{\text{1 yd}}^3}\right)\text{and}\left(\frac{\text{1728 in}{\text{.}}^3}{{\text{1 ft}}^3}\right)$.

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

Hence,

$1\cancel{{\text{ yd}}^3}\times \frac{27\cancel{{\text{ft}}^3}}{1\cancel{{\text{yd}}^3}}\times \frac{\text{1728 in}{\text{.}}^3}{1\cancel{{\text{ft}}^3}}=4.6656\times {10}^4{\text{ in}}^3$

c) Square centimeters (${\text{cm}}^2$) in 1 square foot (${\text{ft}}^2$).

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

$\begin{array}{c}1\text{ ft}=12\text{ in}\text{.}\\ {\left(\text{1ft}\right)}^2={\left(\text{12 in}\right)}^2\\ 1{\text{ ft}}^2=144{\text{ in}}^2\end{array}$

And

$\begin{array}{c}1\text{ in}\text{.}=2.54\text{ cm}\\ {\left(\text{1 in}\text{.}\right)}^2={\left(\text{2}\text{.54 cm}\right)}^2\\ 1{\text{ in}}^2=6.45{\text{ cm}}^2\end{array}$

Thus, correct conversion factors will be $\left(\frac{144\text{ in}{\text{.}}^2}{{\text{1 ft}}^2}\right)\text{and}\left(\frac{\text{6}{\text{.45 cm}}^2}{{\text{1 in}}^2}\right)$.

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

Hence,

$1\cancel{{\text{ ft}}^2}\times \frac{144\cancel{{\text{in}}^2}}{1\cancel{{\text{ft}}^2}}\times \frac{\text{6}{\text{.45 cm}}^2}{1\cancel{{\text{in}}^2}}=9.28\times {10}^2{\text{ cm}}^2$