A two-component model used to determine percent body fat in...

A two-component model used to determine percent body fat in a human body assumes that a fraction $f(<1)$ of the body's total mass $m$ is composed of fat with a density of $0.90 \mathrm{g} / \mathrm{cm}^{3},$ and that the remaining mass of the body is composed of fat-free tissue with a density of 1.10 $\mathrm{g} / \mathrm{cm}^{3} .$ If the specific gravity of the entire body's density is $X,$ show that the percent body fat $(=f \times 100)$ is given by $$\% \text { Body fat }=\frac{495}{X}-450$$

A two-component model used to determine percent body
fat in a human body assumes that a fraction $f(<1)$ of the
body's total mass $m$ is composed of fat with a density of
$0.90 \mathrm{g} / \mathrm{cm}^{3},$ and that the remaining mass of the body is
composed of fat-free tissue with a density of 1.10 $\mathrm{g} / \mathrm{cm}^{3} .$ If
the specific gravity of the entire body's density is $X,$ show
that the percent body fat $(=f \times 100)$ is given by
$$\% \text { Body fat }=\frac{495}{X}-450$$

Key Concepts

Two-Component Body Composition Model

This concept involves partitioning the human body into two distinct components—fat tissue and fat-free tissue—to simplify the estimation of overall body composition. By assuming that each of these components has a uniform density, the model facilitates the derivation of relationships between mass, volume, and density, which can then be used to estimate percent body fat.

Density and Specific Gravity

Density is the mass per unit volume of a substance, and specific gravity is a dimensionless quantity that compares the density of a substance to a reference density (commonly that of water). In the context of body composition, knowing the densities of fat and fat-free tissue allows one to determine the overall density of the body and, consequently, its specific gravity, which is critical for calculating body fat percentage.

Mixture Theory and Mass Fraction Calculation

Mixture theory in this context refers to the process of deriving the overall density or volume of a composite system from its individual components. By considering the mass fraction of fat and the complementary fraction of fat-free tissue, one can form an equation that relates these fractions with the respective densities to obtain an expression for the overall body density (or specific gravity). This forms the basis for algebraic manipulation to express percent body fat in terms of the measured specific gravity.

Transcript

00:01 We're essentially trying to prove the percent body fat equation.

00:04 We're going to use the definition of density and specific gravity, and then we're going to call a fraction f the fat fraction.

00:13 And so we can say that first, the mass of the fat would be equalling m times f, the total mass times the fat fraction.

00:22 This would be equal to the volume of fat times the density of fat.

00:26 We can then say that the mass of the fat free portion would be equalling the total mass times one minus the fat fraction.

00:37 And this would be equalling to the volume of fat free times the density of fat free.

00:47 We can then say that the density of the body is equaling x, which we know that x would be the entire body's density times the density of water.

01:02 And we can then say that this would be equal to the mass, the total mass, divided by the total volume.

01:12 And so this would be equaling the mass of the fat plus the mass of the fat free.

01:20 This would be divided by the volume of the fat plus the volume of the fat free.

01:28 And so we can then say that the density of the body is going to be equaling.

01:38 We can say the total mass m divided by the mass times the fat fraction f divided by the density of fat plus m multiplied by one minus the fat fraction f divided by the density of fat three and so we can then simplify this this is going to be equaling to one divided by the fat fraction divided by the density of fat three plus one minus the fat fraction divided by the density of the fat three.

02:28 Of course, here we're just canceling out all the ms.

02:32 And so we can then say that the fat fraction solving for the fat fraction f, this is going to be equalling the density of fat times the density of fat three, divided by x the entire body's density times the density of water, multiplied by the density of fat -free minus the density of fat.

02:59 This would be minus also the density of fat divided by the density of fat -free minus the density of fat...

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