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1.9 Mathematical Modeling

1.9 Mathematical Modeling. Fitting models to Data using…. Direct Variation Inverse Variation Varies Jointly Least Squares Regression. Direct Variation. Two basic types of linear models. The more general model has a y-intercept that is nonzero; y = mx + b, b≠0 The simpler model is

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1.9 Mathematical Modeling

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  1. 1.9 Mathematical Modeling

  2. Fitting models to Data using…. • Direct Variation • Inverse Variation • Varies Jointly • Least Squares Regression

  3. Direct Variation • Two basic types of linear models. The more general model has a y-intercept that is nonzero; y = mx + b, b≠0 • The simpler model is • y = kx “ y is said to vary directly as x” or be directly proportional to x

  4. Direct Variation • The following statements are equivalent • 1.) yvaries directly as x • 2.) yis directly proportional to x. • 3.) y = kxfor some nonzero constant k. • k is the constant of variation or the constant of proportionality.

  5. Direct Variation Example • In North Carolina, the state income tax is directly proportional to gross income. You are working in NC and your state income tax deduction is $56 for a gross monthly income of $1726. Find a mathematical model that give the NC state income tax in terms of gross income. • y = kx • t = kg • 56 = k(1726) • k= .0324 • So, the equation (or model) for state income tax in NC is • t= .0324g • In other words NC has a state income tax rate of 3.2%

  6. Direct Variation as an nth power • Example: area of a circle, • “the area is directly proportional to the square of the radius r” • Ex. The distance a ball rolls down an inclined plane is directly proportional to the square of the time it rolls. • d = kt² • Ex. T varies directly as the cube of e • t = ke³

  7. Inverse Variation • Equivalent statements; • 1.) yvaries inversely as x • 2.) yis inversely proportional to x • 3.) y = k/x for some constant k

  8. Example • A gas law states that the volume of an enclosed gas varies directly as the temperature and inversely as the pressure. The pressure of a gas is .87kg per square cm when the temperature is 289k and the volume is 8000 cm³ • Write an equation relating pressure, temperature and volume • V= 24.08t/p • Find the pressure when the temp is 298k and the volume is 6988 cm³. • p= .97

  9. JointVariation • To describe two different direct variations in the same statement, the word jointly is used. • Following statements are equivalent • z varies jointly as x and y • z is jointly proportional to x and y • z = kxy for some constant k. • Example: z varies jointly as the nth power of x and the mth power of y. • z = kx˄ny˄m

  10. Example: joint variation • The simple interest for a certain savings account is jointly proportional to the time and the principal. After one quarter (3 months), the interest on a principal of $5000 is $43.75. • Write an equation relating the interest, principal and time. • I = ktp • I = .035tp • Find the interest after 3 quarters. • I= $131.25

  11. Least Squares Regression • Best fitting linear model is called least squares regression. • It is the sum of the squares of actual data values and model values. • Correlation coefficient gives the measure of how well the model fits the data. The closer the value of ӀrӀ is to 1, the better the fit.

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