Direct Variation
May 2012
Learning Objective
The student will learn to solve problems
involving Direct Variation.
Water Pressure
As scuba divers go
deeper under the
water’s surface, they
experience increasing
pressure on their
bodies. The table to
the right depicts the
relationship between
depth and pressure.
𝒚
𝒌𝑷𝒂/𝒎
𝒙
Depth
x (m)
Pressure
y (kPa)
3
29.4
9.8
6
58.8
9.8
9
88.2
9.8
12
117.6
9.8
Direct Variation
Notice that the ratio of the pressure to the
depth is constant.
The pressure is said to vary directly with the
water pressure. This relationship is given by the
following equation:
𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 9.8 × 𝐷𝑒𝑝𝑡ℎ
9.8 is the constant of variation.
Definition of Direct Variation
A linear function defined by an equation of
the form,
𝑦 = 𝑚𝑥 𝑚 ≠ 0
is called a direct variation, and we say that
y varies directly as x. The constant m is
called the constant of variation.
Example: Finding m
Suppose y varies directly as x, and 𝑦 = 15 when
𝑥 = 24.
Consequently,
and
𝑦 = 𝑚𝑥.
15 = 𝑚 × 24

15
24
𝑚=
=
5
8
Example: Finding x
Using the previous example,
𝑦 = 𝑚𝑥.
Where,
𝑚=
15
24
=
5
8
If , 𝑦 = 25 find x.
25 =
8
5
5
𝑥
8
𝑥 = ∙ 25 = 40
Class work:
Page 354,
Oral Exercises 1-4
Homework:
Page 354,
Written Exercises 1-19 odd
Proportion
Stretched Spring
y
y
Unloaded
Loaded
Loaded and stretched
Example: Loaded Spring
The stretch in a loaded spring varies directly with
the load it supports (within the spring’s elastic
limit).
A load of 8 g stretches a certain spring 9.6 cm.
a. Find the constant of variation (the spring
constant) and the equation of the direct
variation.
b. What load would stretch the spring 6 cm?
Example: Loaded Spring
Equation of direct variation:
𝑦 = 𝑘𝑥
where x is the load in grams, y is the resulting
stretch in cm, and k is the spring constant.
a. Substitute 𝑦 = 9.6 𝑐𝑚 when 𝑥 = 8 𝑔.
9.6 = 𝑘 ∙ 8
9.6
𝑘=
= 1.2 𝑐𝑚 𝑔
8
Example: Loaded Spring
Use the equation, 𝑦 = 1.2𝑥 to find 𝑥 when 𝑦 = 6.
b. Substitute 𝑦 = 6 𝑐𝑚,
6 = 1.2𝑥
6
𝑥=
=5𝑔
1.2
Equality of Ratios
The graph of 𝑦 = 𝑚𝑥
is a straight line that
passes through the
origin with slope m.
𝑥2 , 𝑦2
𝑥1 , 𝑦1
O
If neither 𝑥1 nor 𝑥2 is zero, then
𝑦1
𝑥1
=𝑚
Therefore
and
𝑦2
𝑥2
=𝑚
𝑦1
𝑥1
𝑦2
𝑥2
=
Proportion
Such an equality of ratios is called a proportion.
In a direct variation, y is often said to be directly
proportional to x.
The constant of variation, m, is called the
constant of proportionality.
Proportions
The proportion is sometimes written,
means
𝑦1 : 𝑥1 = 𝑦2 : 𝑥2
This is read, “𝑦1 is to 𝑥1 as 𝑦2 is to 𝑥2 ”
Proportions
The equation for a proportion is,
𝑦1 𝑦2
=
𝑥1 𝑥2
If we multiply both sides by 𝑥1 𝑥2 we get,
𝑦1 𝑥2 = 𝑦2 𝑥1
In any proportion, the product of the extremes
equals the product of the means.
Example: Direct Proportionality
The electrical resistance in Ohms () of a wire is
directly proportional to its length:
𝑅1
𝑙1
=
𝑅2
𝑙2
,
Where R is the resistance and l is the length.
If a wire 110 cm long has a resistance of 7.5 ,
what length wire will have a resistance of 12 ?
Electrical Resistance
Let l be the required length in centimeters.
Then
7.5 12
=
110
𝑙
Solving for l gives,
110
𝑙=
12 = 176
7.5
The wire’s length is 176 cm.
Nonlinear Direct Variations
An important equation is physics is,
𝐸=
1
𝑚𝑣 2 ,
2
where E is energy, m is mass, and v is velocity,
E is said to vary directly as m and directly as v2.
Nonlinear Direct Variations
The period of a pendulum is given by,
𝜏=
2𝜋
𝑔
𝑙,
where  is period, 𝑔 is acceleration due to
gravity, and l is length of the pendulum.
The period ()is said to vary directly as the
square root of the length 𝑙 .
Class work:
Page 354,
Oral Exercises 5-12
Homework:
Page 356,
Problems 1-19 odd
Page 357: Mixed Review