Download Section 6.5 – Variation I. Direct Variation II. Inverse Variation

Professor Valdez
Math 203 – Intermediate Algebra
Section 6.5 – Variation
I. Direct Variation
ie, The circumference of a circle C varies directly (or is
directly proportional to) the diameter d because C = π d
where π is the constant of proportionality.
By looking at the graph of this relationship, we see that
this is line passing through the origin with slope of π .
Notice as d increases, C increases. Similarly, as d
decreases, C decreases.
In general, for the direct variation equation y = kx as x
increases, y increases. Similarly, as x decreases, y
decreases.
Example 1 Suppose y varies directly as x, and y = 16 when x = 20. Find the constant of
variation and the variation equation.
II. Inverse Variation
ie, In looking at the equation d = rt and solving for t, we see that t =
d
.
r
If d is constant, the time it takes to travel a specific distance varies inversely (or is inversely
proportional) to the speed r.
8
. We can look at the graph of this by plotting points and examining
r
how time travel times t varies at different speeds r.
If d is 8 miles, then t =
Notice as r increases, t decreases. Similarly, as r decreases, t increases.
In general, for the inverse variation equation y =
k
as x increases, y decreases. Similarly, as
x
x decreases, y increases.
Example 2 Suppose y varies inversely as x, and y = 125 when x = 2/5. Find the constant of
variation and the variation equation.
II. Joint Variation
ie, In looking at the simple interest equation I = prt , if principle value p is fixed then we see
that the interest I is joint proportional to the interest rate r and the time t in years.
Example 3 Suppose y varies jointly with x and z and y = 150 when x = 15 and z = 20. Find
the constant of variation and the variation equation.
III. Combined Variation
Example 4 Suppose w varies jointly as x and y and inversely as square of z.
a. Write a variation equation.
b. If w = 8 when x = 2, y = 6 and z = 4, find the constant of variation.
c. Using the previous parts, find w when x = 2, y = 3, and z = 4.
IV. Applications of Variation
In solving variation problems it’s good to have a strategy.
a. Write a variation equation given a description and creates a table of values.
b. Find the constant of proportionality given values of the variables and restate your
equation.
c. If necessary, find the specified quantity asked for using your equation
Example 5 In a clinical class a student nurse was told that the amount A of the drug
gentamicin administered to patients was directly proportional to the patients weight M (in
kilograms). Suppose 160 mg of gentamicin is administered to a patient who is 40 kg. How
much of the drug should be administered to a patient who weights 55 kg.
Example 6 In photography, the f-stop (the time the opening that lets light into the camera)
for a particular lens is inversely proportional to the aperture (maximum diameter of the
lens). For a particular lens, an f-stop of 2 corresponds to an aperture diameter of 25 mm.
Find the f stop for this lens when the aperture diameter is 12.5 mm.
Example 7 Kinetic energy is the energy associated with motion. The kinetic energy E of an
object is jointly proportional to its mass m and the square of the velocity v. A 0.142 kg
baseball has 113.6 Joules (J) of kinetic energy when its velocity is 40 m/sec. What is the
kinetic energy of the same baseball when its velocity is 20 m/sec?
Example 8 The body mass index (BMI) is used by physicians to determine a patient’s total
body fat. The BMI varies directly as a person’s weight (in pounds) and varies inversely as the
square of the person’s height (in inches). A patient who weights 165 lbs and is 70 in tall has
a BMI of approximately 23. To the nearest whole number, approximate the BMI of a patient
who weights 120 lb with a height of 65 in.