2.8 Variation
Often scientific phenomena are described verbally and need to be translated as
mathematic statements.
The statement:
“ X varies directly with (is directly proportional to) Y.”
... leads to the equation:
X=kY
The value k is called a proportion constant.
It is required only once for a variation (proportion) equation.
For values which vary directly, an increase in one causes an
increase in the other.
ex) M varies directly with the cube root of R.
ex) For an object in free fall, the distance it falls, D , varies directly
(or is directly proportional) to the square of the number of
seconds, T , since it was dropped. It takes 3 seconds for an object
to fall a distance of 144 feet. How far will the object fall after 6
seconds?
X varies inversely with (is inversely proportional to) Y.
k
X=
Y
For values which vary inversely, an increase in one causes a
decrease in the other.
ex) F varies inversely with the fifth power of P.
ex) In kick boxing, the force, F , needed to break a board varies
inversely with the length, L , of the board. If it takes 5 lbs of
pressure to break a board 2 feet long, how much force is needed
to break a board 6 feet long?
X varies jointly with (is jointly proportional to) Y and Z.
X = kYZ
No matter how many variables are used after the proportion
statement, you only need to use one value of k for the equation.
ex)
H varies jointly with W and the square of Z and varies inversely
with the cube of M.
H = 20 when W = 6, Z = 2 and M = 3.
H = ? when W = 2, Z = 1 and M = 4.