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

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Section 3-4 Section 3-4 Direct and Inverse Direct and Inverse Variation Variation Goals Goals Study direct variation Study direct variation Study inverse variation Study inverse variation Study joint variation Study joint variation Study combined variation Study combined variation Solve applied variation problems Solve applied variation problems

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

Page 1: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Section 3-4Section 3-4

Direct and Inverse Direct and Inverse

VariationVariation• GoalsGoals

Study direct variation Study direct variation Study inverse variationStudy inverse variation Study joint variationStudy joint variation Study combined variationStudy combined variation Solve applied variation problemsSolve applied variation problems

Page 2: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Direct VariationDirect Variation• If a car is traveling at a constant rate of 50 If a car is traveling at a constant rate of 50

miles per hour, then the distance miles per hour, then the distance dd traveled in traveled in tt hours is hours is d =d = 50 50tt..• As As tt gets larger, gets larger, dd also gets larger. also gets larger.

• As As tt gets smaller, gets smaller, dd also gets smaller. also gets smaller.

• We say that We say that dd is is directly proportionaldirectly proportional to to tt or or dd varies directlyvaries directly as as tt..

• The number 50 is called the The number 50 is called the constant of constant of variationvariation or the or the constant of proportionalityconstant of proportionality..

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

Direct Variation, Direct Variation,

cont’dcont’d

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

Example 17Example 17

• Suppose Suppose yy varies directly as the varies directly as the square of square of xx..

• Find the constant of variation if Find the constant of variation if yy is 16 is 16 when when xx = 2 and use it to write an = 2 and use it to write an equation of variation.equation of variation.

Page 5: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 17, cont’dExample 17, cont’d

• Solution: Since Solution: Since yy varies directly as the varies directly as the square of square of xx, the general equation will , the general equation will be: be:

• Use the given values of Use the given values of xx and and yy and and solve the equation for solve the equation for kk..

2y kx

Page 6: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 17, cont’dExample 17, cont’d

• Solution, cont’d: The constant of variation is Solution, cont’d: The constant of variation is kk = 4. = 4.

• The variation equation is: The variation equation is: 24y x

Page 7: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Inverse VariationInverse Variation

• Boyle’s law for the expansion of gas is Boyle’s law for the expansion of gas is , where , where VV is the volume of is the volume of the gas, the gas, PP is the pressure, and is the pressure, and KK is a is a constant.constant.• As As PP gets larger, gets larger, VV gets smaller. gets smaller.

• As As PP gets smaller, gets smaller, VV gets larger. gets larger.

• We say that We say that VV is is inversely proportionalinversely proportional to to PP or or VV varies inverselyvaries inversely as as PP..

KV

P

Page 8: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Inverse Variation, Inverse Variation,

cont’dcont’d

Page 9: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 18Example 18

• Suppose Suppose yy varies inversely as the varies inversely as the square root of square root of xx..

• Find the constant of proportionality if Find the constant of proportionality if yy is 15 when is 15 when xx is 9 and use it to write an is 9 and use it to write an equation of variation.equation of variation.

Page 10: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 18, cont’dExample 18, cont’d

• Solution: Since Solution: Since yy varies inversely as varies inversely as the square root of the square root of xx, the general , the general equation will be:equation will be:

ky

x

Page 11: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 18, cont’dExample 18, cont’d

• Solution, cont’d: Solution, cont’d: Replace Replace yy with with 15 and 15 and xx with 9, with 9, and solve for and solve for kk..

• The equation is The equation is

45y

x

Page 12: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 19Example 19

• Determine whether the variation Determine whether the variation between the variables is direct or between the variables is direct or inverse.inverse.

a)a) The time traveled at a constant speed The time traveled at a constant speed and the distance traveledand the distance traveled

b)b) The weight of a car and its gas mileageThe weight of a car and its gas mileage

c)c) The interest rate and the amount of The interest rate and the amount of interest earned on a savings accountinterest earned on a savings account

Page 13: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 19, cont’dExample 19, cont’d

• Solution:Solution:

a)a) The time traveled at a constant speed The time traveled at a constant speed and the distance traveledand the distance traveled

• The longer you travel at a constant The longer you travel at a constant speed the more distance you will cover.speed the more distance you will cover.

• This is an example of a direct variation.This is an example of a direct variation.

Page 14: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 19, cont’dExample 19, cont’d

• Solution, cont’d:Solution, cont’d:

b)b) The weight of a car and its gas mileageThe weight of a car and its gas mileage

• In general, the heavier the car, the In general, the heavier the car, the lower the miles per gallon for that car.lower the miles per gallon for that car.

• This is an example of inverse variation.This is an example of inverse variation.

Page 15: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 19, cont’dExample 19, cont’d

• Solution, cont’d:Solution, cont’d:

c)c) The interest rate and the amount of The interest rate and the amount of interest earned on a savings account interest earned on a savings account

• The higher the interest rate, the more The higher the interest rate, the more interest you will earn on the account.interest you will earn on the account.

• This is an example of direct variation.This is an example of direct variation.

Page 16: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Other Types of Other Types of

VariationVariation

Page 17: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 20Example 20

• Write the following statements as a Write the following statements as a variation equation.variation equation.

a)a) ww varies jointly as varies jointly as yy and the cube of and the cube of xx..

b)b) xx is directly proportional to is directly proportional to yy and and inversely proportional to inversely proportional to zz..

Page 18: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 20, cont’dExample 20, cont’d

• Solution:Solution:

a)a) ww varies jointly as varies jointly as yy and the cube of and the cube of xx..

• Use the definition for joint variation: Use the definition for joint variation:

3w kyx

Page 19: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 20, cont’dExample 20, cont’d

• Solution, cont’d:Solution, cont’d:

b)b) xx is directly proportional to is directly proportional to yy and and inversely proportional to inversely proportional to zz..

• Use the definition for compound Use the definition for compound variation: variation:

kyx

z

Page 20: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 21Example 21

• Suppose Suppose yy is directly proportional to is directly proportional to xx and inversely proportional to the and inversely proportional to the square root of square root of zz..

a)a) Find the constant of variation if Find the constant of variation if yy is 4 is 4 when when xx is 8 and is 8 and zz is 36 and write an is 36 and write an equation of variation.equation of variation.

b)b) Determine Determine yy when when xx is 5 and is 5 and zz is 16. is 16.

Page 21: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 21, cont’dExample 21, cont’d

• Solution: Since Solution: Since yy is directly is directly proportional to proportional to xx and inversely and inversely proportional to the square root of proportional to the square root of z, z, the general equation is: the general equation is:

kxy

z

Page 22: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 21, cont’dExample 21, cont’d

• Solution, cont’d: Solution, cont’d:

a)a) Use the given values of Use the given values of y,y, x, x, and and zz to solve for to solve for kk..

• The equation is The equation is

3xy

z

Page 23: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 21, cont’dExample 21, cont’d

• Solution, cont’d: Solution, cont’d:

b)b) To determine To determine yy when when xx is 5 and is 5 and zz is 16, is 16, evaluate the equation found in part a.evaluate the equation found in part a.

Page 24: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Solving Applied Solving Applied

ProblemsProblems

Page 25: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 22Example 22

• The distance a car travels at a constant The distance a car travels at a constant speed varies directly as the time it travels.speed varies directly as the time it travels.

• Find the variation formula for the distance Find the variation formula for the distance traveled by a car that traveled 220 miles in traveled by a car that traveled 220 miles in 4 hours at a constant speed.4 hours at a constant speed.

• How many miles will the car travel in 7 How many miles will the car travel in 7 hours at that same constant speed?hours at that same constant speed?

Page 26: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 22, cont’dExample 22, cont’d

• Solution: Let Solution: Let dd = distance traveled and = distance traveled and tt = time. = time.

• The general equation is The general equation is dd = = ktkt..

• Substitute 220 for Substitute 220 for dd and 4 for and 4 for t t and and solve forsolve for k. k.• kk = 55 = 55

• The equation is The equation is dd = 55 = 55t.t.

Page 27: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 22, cont’dExample 22, cont’d

• Solution, cont’d: Since the variation Solution, cont’d: Since the variation equation is equation is dd = 55 = 55t, t, we know the car is we know the car is traveling at a rate of 55 miles per hour.traveling at a rate of 55 miles per hour.

• In 7 hours, the car can travel In 7 hours, the car can travel dd = 55(7) = 55(7) = 385 miles.= 385 miles.

Page 28: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 23Example 23

• Ohm’s law says that the current, Ohm’s law says that the current, II, in a , in a wire varies directly as the wire varies directly as the electromotive force, electromotive force, EE, and inversely , and inversely as the resistance, as the resistance, RR..

• If If II is 11 when is 11 when EE is 110 and is 110 and RR is 10, is 10, find find I I if if EE is 220 is 220 and and RR is 11. is 11.

Page 29: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 23, cont’dExample 23, cont’d

• Solution: Since Solution: Since II varies directly as varies directly as EE and inversely as and inversely as RR, the general , the general equation is:equation is:

• Substitute 11 for Substitute 11 for II, 110 for , 110 for EE, and 10 , and 10 for for RR, and solve for, and solve for k. k.

kEI

R

Page 30: Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study.

Example 23, cont’dExample 23, cont’d

• Solution, cont’d: The variation Solution, cont’d: The variation equation is:equation is:

• To find To find I I if if EE is 220 is 220 and and RR is 11, is 11, evaluate the equationevaluate the equation..

1EI

R