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Chapter 13: Problem 103

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 $$ \% \text { Body fat }=\frac{495}{X}-450 . $$

Short Answer

Expert verified
The percent body fat is given by \(\% \text{ Body Fat} = \frac{495}{X} - 450\).

Step by step solution

01

Understand the Variables

Identify the given variables: \(f\) is the fraction of body mass that is fat, \(m\) is the total body mass, \(X\) is the specific gravity of body density, 0.90 g/cm鲁 is the density of fat, and 1.10 g/cm鲁 is the density of fat-free tissue.
02

Express the Total Body Density

The total density \(\rho\) of the body can be calculated using the formula \(\rho = \frac{m}{V}\), where \(V\) is the volume. We know that the density is weighted between fat and fat-free components: \(\rho = f \times 0.90 + (1-f) \times 1.10\).
03

Define Specific Gravity

Specific gravity \(X\) is the ratio of body density to the density of water (1 g/cm鲁). Thus, \(X\) is equal to the body's overall density \(\rho\). Substitute \(\rho\) from Step 2 into \(X = f \times 0.90 + (1-f) \times 1.10\).
04

Solve for Fraction \(f\)

Rearrange the equation from Step 3 to solve for \(f\). Set \(X = f \times 0.90 + 1.10 - 1.10f\), giving \(X = 1.10 - 0.20f\). Solve for \(f\): \(f = \frac{1.10 - X}{0.20}\).
05

Convert Fraction to Percentage

To find the percent body fat, express \(f\) as a percentage: \(% \text{ Body Fat} = f \times 100 = \frac{1.10 - X}{0.20} \times 100\).
06

Simplify the Expression

Simplify \(\frac{1.10 - X}{0.20} \times 100\) to match the given formula. Multiply by 100: \(\% \text{ Body Fat} = \frac{110}{0.20} - \frac{100X}{0.20}\). This simplifies to \(495 - 5X\). Rearrange to \(\% \text{ Body Fat} = \frac{495}{X} - 450\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

specific gravity
Specific gravity is a term used to describe the ratio of the density of a substance to the density of a reference substance; for bodies, this is usually water. Since the density of water is approximately 1 g/cm鲁, the specific gravity of the human body is essentially equal to its density.
  • This means that if the specific gravity of a body is 1, it has the same density as water.
  • A specific gravity greater than 1 indicates that a body is denser than water, while less than 1 implies it is less dense.
Understanding specific gravity is crucial when it comes to calculating body composition, as it allows us to determine what proportion of our body is made up of fat compared to fat-free tissue.
density of human body
The density of the human body is a blended value, combining both the density of fat and the density of fat-free tissue. To find this, we consider the mass and volume of each component part.
The formula used is based on a combination of fat and fat-free tissue densities:
  • The density of fat is approximately 0.90 g/cm鲁.
  • The density of fat-free tissue is about 1.10 g/cm鲁.

These two values are integrated into an equation to calculate overall body density, which plays a pivotal role in estimating the percentage of body fat.
two-component model
The two-component model is a simple yet effective method for measuring body composition. It divides the body into two primary components: fat and fat-free tissue.
  • Fat-free tissue includes muscle, bones, organs, and other non-fat elements.
  • This model assumes that each of these components has a distinct and constant density, which simplifies calculations.

By focusing on just these two components, the model helps calculate the percentage of body fat in a straightforward manner. Although simplistic, it provides valuable insights for assessing individual body composition.
body density
Body density is key in body composition analysis, serving as a measure of the compactness of an individual's body mass. It is calculated by dividing body mass by body volume.
When we know the densities:
  • Body fat density: 0.90 g/cm鲁
  • Fat-free tissue density: 1.10 g/cm鲁

We can deduce the body's overall density. This density is pivotal in determining the specific gravity and, ultimately, the percentage of body fat in the two-component model.
fat-free tissue
Fat-free tissue refers to everything in the body that is not fat, such as muscles, organs, bones, and fluids. In the two-component model, fat-free tissue is assumed to have a uniform density of 1.10 g/cm鲁.

Highlights of fat-free tissue:
  • Includes all body components except fat.
  • Heavier and denser than fat due to higher mineral content such as bones and muscles.

Understanding fat-free tissue is critical because it significantly influences the calculation of body density and specific gravity. Proper analysis of fat-free tissue provides insight into a person's non-fat body mass.
density of fat
The density of fat reflects its less compact nature compared to fat-free tissue. Fat has a lower density, approximately 0.90 g/cm鲁, which is less than that of water.
  • Fat's density explains why it takes up more space for the same mass compared to denser tissues like muscles.
  • This difference in density is crucial for calculating body fat percentage using the two-component model.

The density of fat provides a baseline for understanding its relative proportion in overall body composition. Recognizing this helps in making accurate estimations of body fat percentage through specific equations.

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