User's Manual

Discover Density Set 012–07192A
16
The result of such an analysis is the discovery that y is proportional to x
2
. In
other words,
mass is proportional to diameter
2
.
Evaluating the constant c is not needed.
An Important Theorem:
If a quantity is proportional to a second quantity, and the first quantity is also
proportional to a third quantity, then the first quantity is proportional to the
product of the second and third quantities.
Applying this theorem to the example at hand makes this concept more
clear.
mass is proportional to height, and
mass is proportional to diameter
2
so
mass is proportional to height times diameter
2
In symbols:
M = CHD
2
where C is a constant of proportionality to be determined.
Solving for C gives
C = M / (HD
2
).
Substituting any correlated set of data from the original data set, such as
D = 2.0 cm, H = 6.0 cm, M = 11.12 g (corresponding to the last cone in
group 1) gives
C = 11.12 g / ((6.0 cm)(2.0 cm)
2
)
= 0.46 (g/cm
3
)
The final equation, relating the mass, diameter, and height of all cones made
of this particular alloy, is
M = 0.46 (g/cm
3
) HD
2