Chapter 1, Problem 3KSP

Determine the density of the following object in $\text{g/c}{\text{m}}^{\text{3}}\text{. A cube with edge length = 0}\text{.750 m and mass = 14}\text{.56 kg}\text{.}$

$\begin{array}{l}(\text{a})\\ \text{0}{\text{.0345 g/cm}}^{\text{3}}\\ \text{(b)}\\ \text{1}{\text{.74 g/cm}}^{\text{3}}\\ \text{(c)}\\ {\text{670 g/cm}}^{\text{3}}\\ \text{(d)}\\ \text{53}{\text{.8 g/cm}}^{\text{3}}\\ \text{(e)}\\ \text{14}{\text{.6 g/cm}}^{\text{3}}\end{array}$

Interpretation Introduction

Interpretation:

The density of agiven object in gram per cubic centimeter (${\text{ g/cm}}^3$)is to be determined.

Concept introduction:

The formula for the volume $\left(V\right)$ of a cube with edge $\left(l\right)$ is shown below:

$V=l^3$

The relationship between density, volume, and mass is as follows:

$d=\frac{m}{V}$

Here, $d$ is the density, $V$ is the volume, and $m$ is the mass of the given sample.

Conversion factors are used to convert units of measurement of the same quantity without changing its value.

Conversion factors are derived from the unity bracket method of unit conversion, which consists of a fraction derived from the equality. The units of the denominator and the numerator are different.

The use of a conversion factor to solve numerical problems is called dimensional analysis.

Answer to Problem 3KSP

Option (a).

Explanation of Solution

Given information:

Thelength of the edge of the cube is $l=\text{0}\text{.750 m}$ and the mass of the cube is $m=14.56\text{ kg}$.

The volume of the cube is given by the expression given below:

$V=l^3$

Substitute the value of the length of the edge of the cube in the above expression as shown below:

$\begin{array}{c}V={\left(\text{0}\text{.750 m}\right)}^3\\ =0.42{\text{ m}}^3\end{array}$

The relation between meter $\left(\text{m}\right)$ and centimeter $\left(\text{cm}\right)$ is as follows:

$1\text{ m}={\text{10}}^2\text{ cm}$

Thus, the conversion factor to obtain volume in $\left(\text{cm}\right)$ from $\left(\text{m}\right)$ is $\left(\frac{{10}^2\text{cm}}{1\text{ m}}\right)$. Takingitscube, the conversion factor now becomes $\left(\frac{{10}^6{\text{cm}}^3}{1{\text{ m}}^3}\right)$.

Multiply the conversion factor with $0.42{\text{ m}}^3$ and cancel the units to get the expression:

$0.42\cancel{{\text{m}}^3}\times \left(\frac{{10}^6{\text{cm}}^3}{1\cancel{{\text{m}}^3}}\right)=0.42\times {10}^6{\text{ cm}}^3$

The calculated volume of the givencube is $0.42\times {10}^6{\text{ cm}}^3$.

The relation between grams $\left(\text{g}\right)$ and kilograms $\left(\text{kg}\right)$ is expressed as follows:

$1\text{ kg}={\text{10}}^3\text{ g}$

Thus, the conversion factor to obtain mass in $\left(\text{g}\right)$ from $\left(\text{kg}\right)$ is $\left(\frac{{10}^3\text{ g}}{1\text{ kg}}\right)$.

Multiply the conversion factor with $14.56\text{ kg}$ (mass of the cube) and cancel the units to get the expression:

$14.56\cancel{\text{kg}}\times \left(\frac{{10}^3\text{g}}{1\cancel{\text{kg}}}\right)=14.56\times {10}^3\text{ g}$

The converted mass of thegiven cube is $14.56\times {10}^3\text{ g}$.

The relationship between density, volume, and mass is as follows:

$d=\frac{m}{V}$

Substitute $14.56\times {10}^3\text{ g}$ for $m$ and $0.42\times {10}^6{\text{ cm}}^3$ for $V$ in the above expression

$\begin{array}{c}d=\left(\frac{14.56\times {10}^3\text{ g}}{0.42\times {10}^6{\text{ cm}}^3}\right)\\ =0.0345{\text{ g/cm}}^3\end{array}$

So, the density is $0.0345{\text{ g/cm}}^3$.

Hence, option (a) is correct.

Reasons for incorrect options:

Option (b) is incorrect because, according to the above calculation, the value $\text{1}{\text{.74 g/cm}}^3$ is not obtained. So, it is a wrong answer.

Option (c) is incorrect because, according to the above calculation, the value ${\text{670 g/cm}}^3$ is not obtained. So, it is a wrong answer.

Option (d) is incorrect because, according to the above calculation, the value $\text{53}{\text{.8 g/cm}}^3$ is not obtained. So, it is a wrong answer.

Option (e) is incorrect because, according to the above calculation, the value $14.6{\text{ g/cm}}^3$ is not obtained. So, it is a wrong answer.

Hence, options (b), (c), (d), and (e) are incorrect.