Chapter 2, Problem 29E

A pond has a surface area of ${\text{1552 m}}^{\text{2}}\text{.}$ Convert this quantity into each of the following units:

$\begin{array}{l}\begin{array}{l}\text{a}{\text{. ft}}^{\text{2}}\hfill \\ \text{b}{\text{. km}}^{\text{2}}\hfill \end{array}\\ \text{c}{\text{. dm}}^{\text{2}}\end{array}$

Interpretation Introduction

Interpretation:

The surface area of the pond, $1552{\text{ m}}^2$ is to be converted into given units.

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 29E

Solution:

(a)

$1.671\times {10}^4{\text{ ft}}^2$

(b)

$1.552\times {10}^{-3}{\text{ km}}^2$

(c)

$1.552\times {10}^5{\text{ dm}}^2$

Explanation of Solution

a) ${\text{1552 m}}^2{\text{ to ft}}^2$.

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

$\begin{array}{c}1\text{ m}=39.37\text{ in}\text{.}\\ {\left(\text{1m}\right)}^2={\left(\text{39}\text{.37 in}\right)}^2\\ 1{\text{ m}}^2=1549.99{\text{ in}}^2\end{array}$

And

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

Thus, correct conversion factors will be $\left(\frac{1550\text{ in}{\text{.}}^2}{{\text{1 m}}^2}\right)\text{and}\left(\frac{{\text{1 ft}}^2}{\text{144 in}{\text{.}}^2}\right)$.

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

Hence,

$1552\cancel{{\text{ m}}^2}\times \frac{1549.99\cancel{{\text{in}}^2}}{1\cancel{{\text{m}}^2}}\times \frac{{\text{1 ft}}^2}{144\cancel{{\text{in}}^2}}=1.6705\times {10}^4{\text{ ft}}^2$

b) ${\text{1552 m}}^2{\text{ to km}}^2$.

To convert ${\text{m}}^2$ to ${\text{km}}^2$, 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 $\left(\frac{{\text{1 km}}^2}{{\text{1000000 m}}^2}\right)$.

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

Hence,

$\text{1552 }\cancel{{\text{m}}^2}\times \frac{{\text{1 km}}^2}{\text{1000000}\cancel{{\text{m}}^2}}=1.552\times {10}^{-3}{\text{ km}}^2$

c) ${\text{1552 m}}^2{\text{ to dm}}^2$.

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

$\begin{array}{c}1\text{ m}=10\text{ dm}\\ {\left(\text{1m}\right)}^2={\left(\text{10 dm}\right)}^2\\ 1{\text{ m}}^2=100{\text{ dm}}^2\end{array}$

Thus, correct conversion factor will be $\left(\frac{{\text{100 dm}}^2}{{\text{1 m}}^2}\right)$.

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

Hence,

$\text{1552 }\cancel{{\text{m}}^2}\times \frac{{\text{100 dm}}^2}{\text{1}\cancel{{\text{m}}^2}}=1.552\times {10}^5{\text{ dm}}^2$