A useful measure of an individual's physical condition is

Chapter 3, Problem 3.11

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A useful measure of an individual's physical condition is the fraction of his or her body that consists of fat. This problem describes a simple technique for estimating this fraction by weighing the individual twice, once in air and once submerged in water. (a) A man has body mass mb = 122.5 kg. If he stands on a scale calibrated to read in newtons, what would the reading be? If he then stands on a scale while he is totally submerged in water at 30C (specific gravity = 0.996) and the scale reads 44.0 N, what is the volume of his body (liters)? (Hint: Recall from Archimedes' principle that the weight of a submerged object equals the weight in air minus the buoyant force on the object, which in turn equals the weight of water displaced by the object. Neglect the buoyant force of air.) What is his body density, Pb (kg/L)? (b) Suppose the body is divided into fat and nonfat components, and that Xf (kilograms of fat/kilograms of total body mass) is the fraction of the total body mass that is fat: Xf = Prove that 1 1 Pb Pnf Xf = 1 1 Pf Pnf where Pb, Pf, and Pnf are the average densities ofthe whole body, the fat component, and the nonfat component, respectively. [Suggestion: Start by labeling the masses (mf and mb) and volumes (Vf and Vb) of the fat component of the body and the whole body, and then write expressions for the three densities in terms of these quantities. Then eliminate volumes algebraically and obtain an expression for mf/mb in terms of the densities. 5 ] (c) If the average specific gravity of body fat is 0.9 and that of nonfat tissue is 1.1, what fraction of the man's body in part (a) consists of fat? (d) The body volume calculated in part (a) includes volumes occupied by gas in the digestive tract, sinuses, and lungs. The sum of the first two volumes is roughly 100 mL and the volume of the SIf you can't work out the proof, take the given formula as valid and proceed to the next part. 67 lungs is roughly 1.2 liters. The mass of the gas is negligible. Use this information to improve your estimate of Xf