Determine the approximate percent increase in waist size...

Determine the approximate percent increase in waist size that occurs when a $155-$ -lb person gains 40.0 lbs 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 $1.0 \mathrm{g} / \mathrm{cm}^{3},$ and the density of fat is 0.918 $\mathrm{g} / \mathrm{cm}^{3} .$

Determine the approximate percent increase in waist size that occurs
when a $155-$ -lb person gains 40.0 lbs 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 $1.0 \mathrm{g} / \mathrm{cm}^{3},$ and the density of fat is 0.918 $\mathrm{g} / \mathrm{cm}^{3} .$

Key Concepts

Density

Density is a fundamental property that relates the mass of a substance to its volume. In contexts involving human physiology and fat, it allows one to convert between changes in mass and the resulting changes in volume by using the relationship density = mass/volume.

Cylinder Model Approximation

The cylinder model is a geometric simplification used to represent complex bodies like humans. By approximating the human body as a cylinder, one can use the volume formula (V = ?r²h) to relate changes in mass (and therefore volume) to changes in the body’s dimensions, such as waist size, assuming height remains constant.

Volume Change Due to Mass Gain

Calculating the change in volume resulting from an increase in mass involves understanding that different substances have different densities. When fat is added, the specific density of fat is used to determine the additional volume, which is then integrated into the overall volume model of the human body.

Geometric Scaling Relationships

Geometric scaling relationships are used to determine how dimensional changes in an object affect its overall volume. For a cylinder, if the volume increases while the height remains constant, the radius (related to the waist size in this context) changes according to the square root of the volume ratio. This relationship is critical for assessing percent increases in measurements like waist size.

Percentage Change Calculation

Calculating the percent increase involves comparing the difference between the initial and final dimensions to the original dimension, then expressing this difference as a percentage. This concept is widely used in many fields to quantify the relative change in any measurable quantity.

Transcript

00:08 Let's calculate the approximate percent increase in waste size.

00:14 We have a 155 pound person.

00:21 Let's change this to grams.

00:26 453 .59 grams.

00:29 It's equivalent to one pounds.

00:32 And one gram is equivalent to one centimeter cubed.

00:37 And this will yield 7 .03 times 10 to the 4th centimeters cubed.

00:46 We can use this to calculate the radius.

00:49 Volume is equal to the pi.

00:56 So the volume is equal to the length times the pi are squared.

01:04 And rearranging this all for the radius here.

01:06 Radius is equal to the square root.

01:11 Volume over l pi the volume 7 .03 times 10 to the fourth centimeters cubed over the length we're told we're assuming this is in approximately a cylinder four feet convert this into centimeters one foot is equal of the 30 .48 centimeters times and solving for the radius here we get a radius of 13 .5 centimeters.

01:51 The circumference of this person is equal to 2 pi r.

01:59 So 2 pi 3 .5 centimeters, which will be equal to to 84 .8 centimeters.

02:32 And this person is going to increase.

02:37 So this is the second part here.

02:39 It's a 40 pounds increase.

02:43 And 453 .59 grams in one pound...

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