Direct and Inverse Variation, Slides of Algebra

Download Direct and Inverse Variation and more Slides Algebra in PDF only on Docsity!

§6.8 Model

by Variation

Review §

 Any QUESTIONS About

• §6.7 → Formulas and Applications of

Rational Equations

 Any QUESTIONS About HomeWork

• §6.7 → HW-

MTH 55

Direct Variation

• Many problems lead to equations of the

form y = kx , where k is a constant. Such

eqns are called equations of variation

• DIRECT VARIATION →

When a situation translates to an

equation described by y = kx , with k a

constant, we say that y varies directly

as x. The equation y = kx is called an

equation of direct variation.

Variation Terminology

• Note that for k > 0, any equation of the form y

= kx indicates that as x increases, y increases

as well

• Synonyms
  • “ y varies as x ,”
  • “ y is directly proportional to x ,”
  • “ y is proportional to x ”
• The Synonym Terms also imply direct variation

and are often used

Example  Direct Variation

• If y varies directly

as x , and y = 3 when

x = 12, then find the

eqn of variation

• SOLUTION: The words

“ y varies directly as x ”

indicate an equation of

the form y = kx :

y = kx ⇒ { } 3 = k ⋅{ 12 }

 Solving for k

k = = =.

 Thus the Equation

of Variation

y x 0 25 x 4

Example  Direct Variation cont.

• Graphing the

Equation of

Variation

x y^

0.^25

=^ ^ Direct Variation Always

produces a

SLANTED LINE

that Passes

Thru the

ORIGIN

Example  Bolt Production

• The number of bolts B that a machine can

make varies directly as the time T that it

operates.

• The machine makes 3288 bolts in 2 hr
• How many bolts can it make in 5 hr 1. Familarize and Translate : The problem

states that we have DIRECT VARIATION

between B and T.

• Thus an equation B = kT applies

Example  Bolt Production cont.

3. Carry Out: (^) B = kT ⇒ 3288 bolts = k ⋅ 2 hr - Solve for k : hr

bolts

hr

bolts k 1644 2

• Thus the Equation

of Variation:

T

hr

bolts B (^) ⋅ 

• If T = 5 hrs:

hr bolts hr

bolts B (^1644) ⋅ 5 = 8220 

• Note that k is a RATE with UNITS

Example  Fluid Statics

(units are lb/ft 3 )

Example  Fluid Statics

• Use k = 5 lb/ft 3 in the Direct Variation

Equation to find the pressure at a depth of

40ft

Example  Inverse Variation

• If y varies inversely as x ,

and y = 30 when x = 20,

find the eqn of variation

• SOLUTION: The words

“ y varies inversely as x ”

indicate an equation of

the form y = k / x :

{ } { 20 }

k

x

k y = ⇒ =

 Solving for k

k = 30 ⋅ 20 = 600

 Thus the Equation

of Variation

x

y

Example  Barn Building

• It takes 56 hours for 25 people

to raise a barn.

• How long would it

take 35 people to

complete the job?

• Assume that all

people are

working at the

same rate.

Example  Barn Building cont.

2. Translate: Since inverse

variation applies use N

k T =

3. Carry Out: Find the Constant of

Inverse Proportionality

N

k T = {^ }^ { 25 workers}

k hrs =

k = 56 ⋅ 25 = 1008 Worker⋅ Hrs

Example  Barn Building cont.

3. Carry Out: The

Eqn of Variation

35 workers

1 008worker •hrs T =

 When N = 35. Find T

N

k T =

T = 28. 8 hrs ≈ 29 hrs

4. Chk: A check might be done by

repeating the computations or by

noting that (28.8)(35) and (56)(25)

are both 1008.