Making Models Using Variation 2.5

Fundamentals

  When scientists talk about a mathematical model for a real-world phenomenon, they often mean an equation that describes the relationship between two quantities.

  For instance, the model may describe how the population of an animal species varies with time or how the pressure of a gas varies as its temperature changes.

 Direct Variation

Direct Variation

  If the quantities x and y are related by an equation y = kx for some constant k ≠ 0, we say that:   y varies directly as x.   y is directly proportional to x.   y is proportional to x.

  The constant k is called the constant of proportionality.

  A line is an example of a direct variation – y=mx+b

Direct Variation

 So, the graph of an equation y = kx that describes direct variation is a line with:   Slope k   y-intercept 0

Direct Variation

  During a thunderstorm, you see the lightning before you hear the thunder because light travels much faster than sound.

  The distance between you and the storm varies directly as the time interval between the lightning and the thunder.

  Suppose that the thunder from a storm 5,400 ft away takes 5 s to reach you.

  Determine the constant of proportionality and write the equation for the variation.

E.g. 1—Direct Variation

  (b) Sketch the graph of this equation.

  What does the constant of proportionality represent?

  (c) If the time interval between the lightning and thunder is now 8 s, how far away is the storm?

Direct Variation

 Let d be the distance from you to the storm and let t be the length of the time interval.

  We are given that d varies directly as t.

  So, d = kt

where k is a constant.

Direct Variation

 To find k (the constant of proportionality), we use the fact that t = 5 when d = 5400.   Substituting these values in the equation,

we get: 5400 = k(5)

E.g. 1—Direct Variation

  Substituting this value of k in the equation for d, we obtain: d = 1080t

as the equation for d as a function of t.

Example (a)

Direct Variation

 The graph of the equation d = 1080t is a line through the origin with slope 1080.

  The constant k = 1080 is the approximate speed of sound (in ft/s).

Direct Variation

 When t = 8, we have:

d = 1080 · 8 = 8640

  So, the storm is 8640 ft ≈ 1.6 mi away.

 Inverse Variation

Inverse Variation

 Another equation that is frequently used in mathematical modeling is

where k is a constant.

Inverse Variation

  If the quantities x and y are related by the equation

for some constant k ≠ 0, we say that:   y is inversely proportional to x.   y varies inversely as x.

Inverse Variation

  Ex 35. Loudness of Sound

  The loudness of sound is inversely proportional to the square of the distance, d from the source of the sound. A person who is 10 ft away from a lawnmower experiences a sound level of 70dB

  How loud is the lawn mower when the person is 100 ft away?

Inverse Variation

  L = loudness   L=k/d2

  Now plug in values to solve k to get the constant of proportionality.

  K=L/d2 = 70/(10)2=70*100=7,000

  Now use k and d to find L:   L=(7000)/(100)2 = .7DB   Since 20dB is a whisper, you would imagine that you

cannot hear this lawn mower at 100 ft

 Joint Variation

Joint Variation

  A physical quantity often depends on more than one other quantity.

  If one quantity is proportional to two or more other quantities, we call this relationship joint variation.

Joint Variation   If the quantities x, y, and z are related by

the equation z = kxy

where k is a nonzero constant, we say that:   z varies jointly as x and y.   z is jointly proportional to x and y.

Joint Variation

  In the sciences, relationships between three or more variables are common.

  Any combination of the different types of proportionality that we have discussed is possible.

  For example, if

we say that z is proportional to x and inversely proportional to y.

Newton’s Law of Gravitation

  Newton’s Law of Gravitation says that:

Two objects with masses m1 and m2 attract each other with a force F that is jointly proportional to their masses and inversely proportional to the square of the distance r between the objects.

  Express the law as an equation.

Newton’s Law of Gravitation

  Using the definitions of joint and inverse variation, and the traditional notation G for the gravitational constant of proportionality, we have:

Gravitational Force

  If m1 and m2 are fixed masses, then the gravitational force between them is: F = C/r2

where C = Gm1m2 is a constant.   So gravitational force is inversely proportional to the

radius squared between the two masses.