Chapter 2, Problem 30E

An orange has a volume of ${\text{54 cm}}^{\text{3}}$. Convert this quantity into each of the following units:

${\text{mm}}^{\text{3}}$

$\text{in}{\text{.}}^{\text{3}}$

${\text{dm}}^{\text{3}}$

Interpretation Introduction

Interpretation:

The volume of the orange, $54{\text{ cm}}^3$, 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 the conversion factor, $\frac{u_2}{u_1}$, then multiply the conversion factor with the given quantity, $a{u}_1$, and cancel the 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 30E

Solution:

(a)

$5.4\times {10}^4{\text{ mm}}^{\text{3}}$

(b)

$3.3\text{ in}{\text{.}}^{\text{3}}$

(c)

$5.4\times {10}^{-2}{\text{ dm}}^{\text{3}}$

Explanation of Solution

a) ${\text{mm}}^{\text{3}}$

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

$\begin{array}{c}1\text{ cm}=10\text{ mm}\\ {\left(\text{1cm}\right)}^3={\left(\text{10 mm}\right)}^3\\ 1{\text{ cm}}^{\text{3}}=1000{\text{ mm}}^{\text{3}}\end{array}$

Thus, the correct conversion factor is $\left(\frac{{\text{1000 mm}}^{\text{3}}}{1{\text{ cm}}^3}\right)$.

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

Hence,

$\left(\text{54 }\cancel{{\text{cm}}^{\text{3}}}\times \frac{1000m{m}^3}{1\cancel{{\text{cm}}^{\text{3}}}}\right)=5.4\times {10}^4{\text{ mm}}^{\text{3}}$

b) $\text{in}{\text{.}}^{\text{3}}$

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

$\begin{array}{c}2.54\text{ cm}=1\text{ in}\text{.}\\ {\left(\text{2}\text{.54 cm}\right)}^3={\left(\text{1in}\text{.}\right)}^3\\ \text{16}{\text{.38 cm}}^{\text{3}}=1\text{ in}{\text{.}}^{\text{3}}\end{array}$

Thus, the correct conversion factor is $\left(\frac{\text{1 in}{\text{.}}^{\text{3}}}{16.38{\text{ cm}}^3}\right)$.

Multiply the conversion factor with the given quantity and cancel the units to get the desired quantity.

Hence,

$\left(\text{54 }\cancel{{\text{cm}}^{\text{3}}}\times \frac{1i{n}^3}{16.38\cancel{{\text{cm}}^{\text{3}}}}\right)=3.3\text{ in}{\text{.}}^{\text{3}}$

c) ${\text{dm}}^{\text{3}}$

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

$\begin{array}{c}\text{10 cm}=1\text{ dm}\\ {\left(\text{10 cm}\right)}^3={\left(\text{1dm}\right)}^3\\ {\text{1000 cm}}^{\text{3}}=1{\text{ dm}}^{\text{3}}\end{array}$

Thus, the correct conversion factor is $\left(\frac{{\text{1 dm}}^{\text{3}}}{1000{\text{ cm}}^3}\right)$.

Multiply the conversion factor with the given quantity and cancel the units to get the desired quantity.

Hence,

$\left(\text{54 }\cancel{{\text{cm}}^{\text{3}}}\times \frac{1d{m}^3}{1000\cancel{{\text{cm}}^{\text{3}}}}\right)=5.4\times {10}^{-2}{\text{ dm}}^{\text{3}}$