Presentation on theme: "Mathematical Modeling"— Presentation transcript:
1 Mathematical Modeling PRECALCULUS IMathematical ModelingDirect, inverse, joint variations; Least squares regressionDr. Claude S. Moore Danville Community College
2 Direct Variation Statements 1. y varies directly as x.2. y is directly proportional to x.3. y = kx for some nonzero constant m.NOTE: k is the constant of variation or the constant of proportionality.Example: If y = 3 when x = 2, find k. y = kx yields 3 = m(2) or m = 1.5.Thus, y = 1.5x.
3 Direct Variation as nth Power 1. y varies directly as the nth power of x.2. y is directly proportional to the nth power of x.3. y = kxn for some nonzero constant k.NOTE: k is the constant of variation or constant of proportionality.
4 Inverse Variation Statements 1. y varies inversely as x.2. y is inversely proportional to x.3. y = k / x for some nonzero constant k.NOTE: k is the constant of variation or the constant of proportionality.Example: If y = 3 when x = 2, find k. y = k / x yields 3 = k / 2 or k = 6.Thus, y = 6 / x.
5 Joint Variation Statements 1. z varies jointly as x and y.2. z is jointly proportional to x and y.3. z = kxy for some nonzero constant k.NOTE: k is the constant of variation. Example: If z = 15 when x = 2 and y = 3, find k. z = kxy yields 15 = k(2)(3) or k = 15/6 = 2.5.Thus, z = 2.5xy.
6 USING DIRECT AND INVERSE VARIATION IN REAL LIFE Writing and Using a ModelBICYCLING A bicyclist tips the bicycle when making turn. The angle B of the bicycle from the vertical direction is called the banking angle.banking angle, B
7 Writing and Using a Model Turning RadiusBanking angle (degrees)Find an inverse variation model that relates B and r.SOLUTIONFrom the graph, you can see that B = 32° when r = 3.5 feet.B =krWrite direct variation model.32 =k3.5Substitute 32 for B and 3.5 for r.112 =kSolve for k.The model is B = , where B is in degrees and r is in feet.112r
8 Writing and Using a Model Turning RadiusBanking angle (degrees)Use the model to find the banking angle for a turning radius of 5 feet.SOLUTIONSubstitute 5 for r in the model you just found.B =1125= 22.4When the turning radius is 5 feet, the banking angle is about 22°.