Example 11-10 Measuring Body Fat
In the human body the density of lean muscle is about 1.06 * 103 kg>m3, and the density of fat tissue is about
9.30 * 102 kg>m3. If a person who weighs 833 N in air has an apparent weight of 27.4 N when submerged in water, what
percent of his body mass is fat? (Assume that the person is made of muscle and fat only.)
Set Up
To determine what fraction of this person’s
body mass is fat, we’ll first determine the
­average density r of his body. To do this, we’ll
first find his volume V using Equation 11-15
and the measured values w = 833 N and
wapp = 27.4 N. Then we can use Equation
11-16 to calculate his density r. We’ll next
compare r to the given densities of muscle and
fat to determine the fat fraction.
Solve
Solve Equation 11-15 for the person’s volume
V, then substitute the given values of wapp,
w, g, and rfluid = 1.000 * 103 kg>m3 (the
density of water).
Two equations for the apparent weight wapp
of an object of weight w, volume V, and
density r submerged in fluid of density rfluid:
wapp = w 2 rfluidVg
wapp = (r 2 rfluid)Vg
(11-15)
(11-16)
fat mass: mfat
muscle mass: mmuscle
total mass: m = mfat + mmuscle
Rearrange Equation 11-15 to solve for V:
rfluid Vg = w - wapp
V =
w - wapp
rfluid g
= 0.0822 m3
=
833 N - 27.4 N
11.000 * 103 kg>m3 2 19.80 m>s 2 2
Rearrange Equation 11-16 to solve for r:
Now use this value of V in Equation 11-16
and solve for the person’s density r. We’ll keep 1r - r 2Vg = w
fluid
app
an extra significant figure for this intermediate
wapp
calculation and adjust them at the end.
r - rfluid =
Vg
wapp
r = rfluid +
Vg
27.4 N
= 1.000 * 103 kg>m3 +
10.0822 m3 2 19.80 m>s 2 2
= 1.034 * 103 kg>m3
The overall density r is less than the density
of lean muscle but greater than the density
of fat, so the body is a mixture of these two
quantities. Use the definition of density,
r = m>V, to calculate the fraction x of the
person’s mass that is fat.
Let x = fraction of body mass m that is fat.
Then the mass of fat in the body is
mfat = xm
The remaining mass in the body is muscle:
mmuscle = m 2 mfat = m 2 xm = (1 2 x)m
Volume of body fat = 1mass of fat2 > 1density of fat2, so
mfat
xm
=
Vfat =
rfat
rfat
Volume of body muscle = 1mass of muscle2 > 1density of muscle2, so
Vmuscle =
11 - x2m
mmuscle
=
rmuscle
rmuscle
Overall volume of body = 1mass of body2 > 1overall density of body2, so
m
V =
r
The total volume of the body equals the sum of the volume of fat and
the volume of muscle:
V = Vfat + Vmuscle , so
11 - x2m
m
xm
+
=
r
rfat
rmuscle
Rearrange this equation and solve for x:
11 - x2
1
x
+
=
r
rfat
rmuscle
1
1
x
x
1
1
=
= xa
b
r
rmuscle
rfat
rmuscle
rfat
rmuscle
x =
a
1
1
b
rfat
rmuscle
1
1
a
b
3
3
1.034 * 10 kg>m
1.06 * 103 kg>m3
a
=
1
1
b
r
rmuscle
1
1
b
9.30 * 102 kg>m3
1.06 * 103 kg>m3
= 0.180
a
Reflect
Our result tells us that 0.180 (or 18.0%) of this
person’s mass is fat. Two-thirds of Americans
have a percent body fat of 25% or more and
are therefore overweight or obese. Adult men
with between 18 and 24% body fat are considered healthy.
We can check our result for the person’s
density by calculating it in a different way:
Take the person’s weight of 833 N and divide
by g to get the mass, then divide that by the
volume that we calculated above from Equation 11-15. Happily, we get the same result as
we did using Equation 11-16.
Note that the method outlined here is
only an approximation of an actual body fat
calculation; we ignored the volumes of the
lungs and bones.
Second calculation of density:
m
r =
V
Determine mass from weight:
w
833 N
= 85.0 kg
w = mg, so m =
=
g
9.80 m>s 2
From above, V = 0.0822 m3. So the person’s density is
85.0 kg
= 1.03 * 103 kg>m3
r =
0.0822 m3