We’re asked to determine the **approximate percent increase** in waist size that occurs when a **150-lb person gains 35.0 lb of fat. **

We will assume that the volume of a person can be modeled by a cylinder that is 4.0 ft tall.

Recall that the **percent increase** is given by the equation:

$\overline{){\mathbf{\%}}{\mathbf{Increase}}{\mathbf{=}}\frac{\mathbf{increase}\mathbf{}\mathbf{in}\mathbf{}\mathbf{waist}\mathbf{}\mathbf{size}}{\mathbf{original}\mathbf{}\mathbf{waist}\mathbf{}\mathbf{size}}}$

Since we're given the density, we can first solve for the waist size of the person to determine the * percent increase* by doing these steps:

*Step 1. Write the equation for the volume of the person using the equation for the volume of a cylinder:*

$\overline{){\mathbf{Volume}}{\mathbf{=}}{{\mathbf{\pi r}}}^{{\mathbf{2}}}{\mathbf{h}}}$

*r = radius*

*π = 3.1416 (constant)*

*h = height *

* *

*Step 2. Determine the change in volume of the person before and after gaining fat based on the given density (of the person and fat) using the equation: *

$\overline{){\mathbf{density}}{\mathbf{=}}\frac{\mathbf{mass}}{\mathbf{volume}}}$

*Step 3. Determine the % increase in waist size, by solving for the change in radius, r. *

We can solve for the waist size using the equation for the circumference *(assuming it is circular)*:

$\overline{){\mathbf{Circumference}}{\mathbf{=}}{\mathbf{2}}{\mathbf{\pi r}}}$

Approximate the percent increase in waist size that occurs when a 150-lb person gains 35.0 lb of fat. Assume that the volume of the person can be modeled by a cylinder that is 4.0 ft tall. The average density of a human is about 1.0 g/cm^{3}, and the density of fat is 0.918 g/cm^{3}.

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