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.

Depth x (m)

Pressure y (kPa)

𝒚

𝒙 𝒌𝑷𝒂/𝒎

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, 𝑦 = 𝑚𝑥.

and 15 = 𝑚 × 24

𝑚 =15

24=

5

8

Example: Finding x

Using the previous example,

𝑦 = 𝑚𝑥.

Where, 𝑚 =15

24=

5

8

If , 𝑦 = 25 find x.

25 =5

8𝑥

𝑥 =8

5∙ 25 = 40

Class work:

Page 354,

Oral Exercises 1-4

Homework:

Page 354,

Written Exercises 1-19 odd

Proportion

Stretched Spring

Unloaded Loaded Loaded and stretched

y

y

Example: Loaded Spring

The stretch in a loaded spring varies directly with the load it supports (within the spring’s elastic limit).

a. Find the constant of variation (the spring constant) and the equation of the direct variation.

A load of 8 g stretches a certain spring 9.6 cm.

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

𝑘 =9.6

8= 1.2 𝑐𝑚 𝑔

Example: Loaded Spring

Use the equation, 𝑦 = 1.2𝑥 to find 𝑥 when 𝑦 = 6.

b. Substitute 𝑦 = 6 𝑐𝑚,

6 = 1.2𝑥

𝑥 =6

1.2= 5 𝑔

Equality of Ratios

The graph of 𝑦 = 𝑚𝑥 is a straight line that passes through the origin with slope m.

O

𝑥1, 𝑦1

𝑥2, 𝑦2

If neither 𝑥1 nor 𝑥2 is zero, then

𝑦1

𝑥1= 𝑚 and

𝑦2

𝑥2= 𝑚

Therefore 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,

𝑦1: 𝑥1 = 𝑦2: 𝑥2

This is read, “𝑦1 is to 𝑥1 as 𝑦2 is to 𝑥2”

means

Proportions

If we multiply both sides by 𝑥1𝑥2 we get,

The equation for a proportion is,

𝑦1𝑥1

=𝑦2𝑥2

𝑦1𝑥2 = 𝑦2𝑥1

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

110=12

𝑙

Solving for l gives,

𝑙 =110

7.512 = 176

The wire’s length is 176 cm.

Nonlinear Direct Variations

An important equation is physics is,

𝐸 =1

2𝑚𝑣2,

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

𝑔𝑙,

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