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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
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 .
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