Answer
13 percent
Work Step by Step
Since \[1\text{ lb}=453.59\text{ g}\], convert the mass of the person into grams as follows:
\[\begin{align}
& m=155\text{ lb}\left( \frac{453.59\text{ g}}{1\text{ lb}} \right) \\
& =7.03\times {{10}^{4}}\text{ g}
\end{align}\]
Now, the volume of the person is calculated as follows:
\[V=\frac{m}{d}\]
Here, m is the mass and d is the density.
Substitute \[7.03\times {{10}^{4}}\text{ g}\] for m and \[1.0\text{ g/c}{{\text{m}}^{\text{3}}}\] for d:
\[\begin{align}
& V=\frac{7.03\times {{10}^{4}}\text{ g}}{1.0\text{ g/c}{{\text{m}}^{\text{3}}}} \\
& =7.03\times \text{1}{{\text{0}}^{4}}\text{ c}{{\text{m}}^{3}}
\end{align}\]
Now, the height of the person is \[4\text{ ft}\]. One foot is equal to 30.48 cm. Thus,
\[\begin{align}
& h=4\text{ ft}\left( \frac{30.48\text{ cm}}{1\text{ ft}} \right) \\
& =121.92\text{ cm}
\end{align}\]
Now, the volume of the cylinder is as follows:
\[V=\pi {{r}^{2}}h\]
Rearrange the equation:
\[{{r}^{2}}=\frac{V}{\pi h}\]
Substitute \[121.92\text{ cm}\] for h and \[7.03\times \text{1}{{\text{0}}^{4}}\text{ c}{{\text{m}}^{3}}\] for V:
\[\begin{align}
& {{r}^{2}}=\frac{7.03\times \text{1}{{\text{0}}^{4}}\text{ c}{{\text{m}}^{3}}}{\left( 3.14 \right)\left( 121.92\text{ cm} \right)} \\
& r=\sqrt{\frac{7.03\times \text{1}{{\text{0}}^{4}}\text{ c}{{\text{m}}^{3}}}{\left( 3.14 \right)\left( 121.92\text{ cm} \right)}} \\
& =13.55\text{ cm}
\end{align}\]
The person is modeled by a cylinder. Thus, the waist of the person is equal to the circumference of the circle. Thus, the waist is calculated as follows:
\[w=2\pi r\]
Substitute \[13.55\text{ cm}\] for r:
\[\begin{align}
& {{w}_{1}}=2\left( 3.14 \right)\left( 13.55\text{ cm} \right) \\
& =85.1\text{ cm}
\end{align}\]
Now, calculate the volume of the fat as follows:
\[\begin{align}
& {{V}_{\text{f}}}=\left( \frac{40.0\text{ lb}}{0.918\text{ g/c}{{\text{m}}^{\text{3}}}} \right)\left( \frac{453.59\text{ g}}{1\text{ lb}} \right) \\
& =1.98\times {{10}^{4}}\text{ c}{{\text{m}}^{3}}
\end{align}\]
The total volume of the person is the sum of the volume of the person and the volume of the fat. Thus,
\[\begin{align}
& {{V}_{\text{T}}}=\left( 7.03\times {{10}^{4}}+1.98\times {{10}^{4}} \right)\text{ c}{{\text{m}}^{3}} \\
& =9.01\times {{10}^{4}}\text{ c}{{\text{m}}^{3}}
\end{align}\]
Now, the radius of cylinder is as follows:
\[\begin{align}
& {{r}^{2}}=\frac{9.01\times \text{1}{{\text{0}}^{4}}\text{ c}{{\text{m}}^{3}}}{\left( 3.14 \right)\left( 121.92\text{ cm} \right)} \\
& r=\sqrt{\frac{9.01\times \text{1}{{\text{0}}^{4}}\text{ c}{{\text{m}}^{3}}}{\left( 3.14 \right)\left( 121.92\text{ cm} \right)}} \\
& =15.34\text{ cm}
\end{align}\]
Now, as the waist of the person is equal to circumference of the circle with radius r,
\[\begin{align}
& {{w}_{2}}=2\left( 3.14 \right)\left( 15.34\text{ cm} \right) \\
& =96.34\text{ cm}
\end{align}\]
Now, the percentage increase in the waist size is calculated as follows:
\[\text{percentage}\,=\frac{{{w}_{2}}-{{w}_{1}}}{{{w}_{1}}}\times 100\]
Substitute \[96.34\text{ cm}\] for \[{{w}_{2}}\] and \[\text{85}\text{.1 cm}\] for \[{{w}_{1}}\]:
\[\begin{align}
& \text{percentage}\,=\frac{96.34\text{ cm}-85.1\text{ cm}}{\text{85}\text{.1 cm}}\times 100 \\
& =13\,\text{percent}
\end{align}\]
The percentage increase in the waist is 13.
