Answer:
x_fat = [ 0.5*(Wsa + Wsw) - p_muscle*V ] / V*( p_fat - p_muscle )
Explanation:
Given:
- The total volume of body = V
- The average density of the body = p_avg
- The density of muscle = p_muscle
- The density of fat = p_fat
Find:
Obtain a relation for the volume fraction of body fat x_fat
Solution:
- The volume of the fat is given by:
V_fat = x_fat*V
- The volume of the muscle is given by:
V_muscle = V - V_fat
= V - x_fat*V
=V*( 1 - x_fat )
- We will use the conservation of mass for the body related as:
mass_fat + mass_muscle = Total average mass
p_fat*V_fat + p_muscle*V_muscle = p_avg*V
p_fat*x_fat*V + p_muscle*V*( 1 - x_fat ) = p_avg*V
p_fat*x_fat + p_muscle*( 1 - x_fat ) = p_avg
- To determine p_1 we weigh the body in air:
Weight reading (Wsa) = m = p_1*V
p_1 = Wsa / V*g
- To determine p_2 we weigh the body in water:
Weight reading (Wsw) = m - p_w*V= p_1*V - p_w*V
Weight reading (Wsw) = V*(p_1 - p_w) = V*(p_2)
Where, p_2 = p_1 - p_water
p_2 = Wsw / V
- The average density p_avg:
p_avg = 0.5*(p_1 + p_2)
p_avg = 0.5*(Wsa / V + Wsw / V)
p_avg = 0.5*(Wsa + Wsw) / V
- Plug in the mass equation:
p_fat*x_fat + p_muscle*( 1 - x_fat ) = 0.5*(Wsa + Wsw) / V
x_fat*( p_fat - p_muscle ) = 0.5*(Wsa + Wsw) / V - p_muscle
x_fat = [ 0.5*(Wsa + Wsw) - p_muscle*V ] / V*( p_fat - p_muscle )
