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# A scale under water

## Problem

Normal bathroom scales only give the weight of a person. This does not give one enough information to determine the body fat percentage of a person. A technique called “hydrostatic weighing” or “underwater weighing” can determine a person’s density, and therefore give information about the percent of body fat in a person’s body.
To perform hydrostatic weighing, a person is first weighed in the air while standing on a regular scale. Then the person is weighed while underwater with the air expelled from their lungs. The measured weight of a person underwater will be less than the measured weight of the person in the air because of the buoyant force acting on the person.
Figure 1. The apparatus involved in hydrostatic weighing is depicted.
Once the weights in air and water are determined, one can calculate the average density of a person. Knowing the average density of the person will allow an estimate of the body fat percentage since fat is less dense than muscle and bone. People with a high average body density have more muscle per weight and so have a smaller body fat percentage.
Consider the data below that was taken for someone using the technique of underwater weighing. (The density of the water used was 1000 kg/mcubed) Assume the acceleration due to gravity is 10m/ssquared. All measurements were taken at a depth of 5m.
PersonMeasured weight in airMeasured weight in water
A652 Newtons36.0 Newtons
B713 Newtons98.0 Newtons
C684 Newtons54.0 Newtons
D597 Newtons89.0 Newtons
What is the volume of person C?