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College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson. Coordinates and Graphs. 2. Making Model Using Variation. 2.5. Fundamentals.
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College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson
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.
Fundamentals • In this section, we study a kind of modeling called variation.
Modeling Variation • Two types of mathematical models occur so often that they are given special names.
Direct Variation • Direct variation occurs when one quantity is a constant multiple of the other. • So, we use an equation of the form y = kxto model this dependence.
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 asx. • y is directly proportional tox. • y is proportional tox. • The constant k is called the constant of proportionality.
Direct Variation • Recall that the graph of an equation of the form y = mx + b is a line with: • Slope m • y-intercept b
Direct Variation • So, the graph of an equation y = kx that describes direct variation is a line with: • Slope k • y-intercept 0
E.g. 1—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.
E.g. 1—Direct Variation • 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?
Example (a) E.g. 1—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.
Example (a) E.g. 1—Direct Variation • To find k, we use the fact that t = 5 when d = 5400. • Substituting these values in the equation, we get: 5400 = k(5)
Example (a) E.g. 1—Direct Variation • Substituting this value of k in the equation for d, we obtain: d = 1080tas the equation for d as a function of t.
Example (b) E.g. 1—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).
Example (c) E.g. 1—Direct Variation • When t = 8, we have: • d = 1080 ∙ 8 = 8640 • So, the storm is 8640 ft ≈ 1.6 mi away.
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 equationfor some constant k ≠ 0, we say that: • y is inversely proportional tox. • y varies inversely asx.
Inverse Variation • The graph of y = k/x for x > 0 is shown for the case k > 0. • It gives a picture of what happens when y is inversely proportional to x.
E.g. 2—Inverse Variation • Boyle’s Law states that: • When a sample of gas is compressed at a constant temperature, the pressure of the gas is inversely proportional to the volume of the gas.
E.g. 2—Inverse Variation • Suppose the pressure of a sample of air that occupies 0.106 m3 at 25°C is 50 kPa. • Find the constant of proportionality. • Write the equation that expresses the inverse proportionality.
E.g. 2—Inverse Variation • If the sample expands to a volume of 0.3 m3, find the new pressure.
Example (a) E.g. 2—Inverse Variation • Let P be the pressure of the sample of gas and let V be its volume. • Then, by the definition of inverse proportionality, we have: where k is a constant.
Example (a) E.g. 2—Inverse Variation • To find k,we use the fact that P = 50 when V = 0.106. • Substituting these values in the equation, we get:k = (50)(0.106) = 5.3
Example (a) E.g. 2—Inverse Variation • Putting this value of k in the equation for P, we have:
Example (b) E.g. 2—Inverse Variation • When V = 0.3, we have: • So, the new pressure is about 17.7 kPa.
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 = kxywhere k is a nonzero constant, we say that: • z varies jointly asx and y. • z is jointly proportional tox 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 tox and inversely proportional to y.
E.g. 3—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.
E.g. 3—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/r2where C = Gm1m2 is a constant.
Gravitational Force • The figure shows the graph of this equation for r > 0 with C = 1. • Observe how the gravitational attraction decreases with increasing distance.
