Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study

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Text of Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation...

  • Section 3-4Direct and Inverse VariationGoalsStudy direct variation Study inverse variationStudy joint variationStudy combined variationSolve applied variation problems

  • Direct VariationIf a car is traveling at a constant rate of 50 miles per hour, then the distance d traveled in t hours is d = 50t.As t gets larger, d also gets larger.As t gets smaller, d also gets smaller.We say that d is directly proportional to t or d varies directly as t.The number 50 is called the constant of variation or the constant of proportionality.

  • Direct Variation, contd

  • Example 17Suppose y varies directly as the square of x.Find the constant of variation if y is 16 when x = 2 and use it to write an equation of variation.

  • Example 17, contdSolution: Since y varies directly as the square of x, the general equation will be: Use the given values of x and y and solve the equation for k.

  • Example 17, contdSolution, contd: The constant of variation is k = 4.The variation equation is:

  • Inverse VariationBoyles law for the expansion of gas is , where V is the volume of the gas, P is the pressure, and K is a constant.As P gets larger, V gets smaller.As P gets smaller, V gets larger.We say that V is inversely proportional to P or V varies inversely as P.

  • Inverse Variation, contd

  • Example 18Suppose y varies inversely as the square root of x.Find the constant of proportionality if y is 15 when x is 9 and use it to write an equation of variation.

  • Example 18, contdSolution: Since y varies inversely as the square root of x, the general equation will be:

  • Example 18, contdSolution, contd: Replace y with 15 and x with 9, and solve for k.The equation is

  • Example 19Determine whether the variation between the variables is direct or inverse.The time traveled at a constant speed and the distance traveledThe weight of a car and its gas mileageThe interest rate and the amount of interest earned on a savings account

  • Example 19, contdSolution:The time traveled at a constant speed and the distance traveledThe longer you travel at a constant speed the more distance you will cover.This is an example of a direct variation.

  • Example 19, contdSolution, contd:The weight of a car and its gas mileageIn general, the heavier the car, the lower the miles per gallon for that car.This is an example of inverse variation.

  • Example 19, contdSolution, contd:The interest rate and the amount of interest earned on a savings account The higher the interest rate, the more interest you will earn on the account.This is an example of direct variation.

  • Other Types of Variation

  • Example 20Write the following statements as a variation equation. w varies jointly as y and the cube of x. x is directly proportional to y and inversely proportional to z.

  • Example 20, contdSolution: w varies jointly as y and the cube of x.Use the definition for joint variation:

  • Example 20, contdSolution, contd: x is directly proportional to y and inversely proportional to z.Use the definition for compound variation:

  • Example 21Suppose y is directly proportional to x and inversely proportional to the square root of z.Find the constant of variation if y is 4 when x is 8 and z is 36 and write an equation of variation.Determine y when x is 5 and z is 16.

  • Example 21, contdSolution: Since y is directly proportional to x and inversely proportional to the square root of z, the general equation is:

  • Example 21, contdSolution, contd: Use the given values of y, x, and z to solve for k.

    The equation is

  • Example 21, contdSolution, contd: To determine y when x is 5 and z is 16, evaluate the equation found in part a.

  • Solving Applied Problems

  • Example 22The distance a car travels at a constant speed varies directly as the time it travels.Find the variation formula for the distance traveled by a car that traveled 220 miles in 4 hours at a constant speed.How many miles will the car travel in 7 hours at that same constant speed?

  • Example 22, contdSolution: Let d = distance traveled and t = time.The general equation is d = kt.Substitute 220 for d and 4 for t and solve for k. k = 55The equation is d = 55t.

  • Example 22, contdSolution, contd: Since the variation equation is d = 55t, we know the car is traveling at a rate of 55 miles per hour.In 7 hours, the car can travel d = 55(7) = 385 miles.

  • Example 23Ohms law says that the current, I, in a wire varies directly as the electromotive force, E, and inversely as the resistance, R.If I is 11 when E is 110 and R is 10, find I if E is 220 and R is 11.

  • Example 23, contdSolution: Since I varies directly as E and inversely as R, the general equation is:Substitute 11 for I, 110 for E, and 10 for R, and solve for k.

  • Example 23, contdSolution, contd: The variation equation is:To find I if E is 220 and R is 11, evaluate the equation.

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