[email protected] MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 1 Bruce Mayer, PE Chabot...

[email protected] • MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§6.8 Model§6.8 Modelby Variationby Variation



Review §Review §

Any QUESTIONS About• §6.7 → Formulas and Applications of

Rational Equations

Any QUESTIONS About HomeWork• §6.7 → HW-28

6.7 MTH 55



§6.8 Direct and Inverse Variation§6.8 Direct and Inverse Variation

Equations of Direct Variation

Problem Solving with Direction Variation

Equations of Inverse Variation Problem Solving with

Inverse Variation



Direct VariationDirect Variation

Many problems lead to equations of the form y = kx, where k is a constant. Such eqns are called equations of variation

DIRECT VARIATIONDIRECT 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 TerminologyVariation 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



The Constant “The Constant “kk””

For the Direct Variation Equation kxy

The constant k is called the constant of proportionality or the variation constant.

k can be found if one pair of values for x and y is known.

Once k is known, other (x,y) pairs can be determined



Example Example Direct Variation 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: 123 kkxy

Solving for k

2504

1

12

3.k

Thus the Equation of Variation

xxy 2504

1.



Example Example Direct Variation Direct Variation cont. cont.

xy 250.

Graphing the Equation of Variation

Direct Variation Always produces a SLANTED LINE that Passes Thru the ORIGIN



Example Example Direct Variation Direct Variation Find an equation in

which a varies directly as b, and a = 15 when b = 25.

Find the value of a when b = 36

SOLUTION:

Thus the Variation Eqnba 60.

Sub b = 36 into Eqn

2515 kkba

605

3

25

15.k

5

3216213660 ..a

Thus when b = 36, then the value of a is 21-3/5



Example Example Bolt Production 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 Example Bolt Production Bolt Production cont.1 cont.1

3. Carry Out: hrkboltskTB 23288

• Solve for k:hr

bolts

hr

boltsk 1644

2

3288

• Thus the Equation of Variation:

Thr

boltsB

1644

• If T = 5 hrs:

boltshrhr

boltsB 822051644

• Note that k is a RATE with UNITS



Example Example Fluid Statics Fluid Statics

The pressure exerted by a liquid at given point varies directly as the depth of the point beneath the surface of the liquid.

If a certain liquid exerts a pressure of 50 pounds per square foot (psf) at a depth of 10 feet, then find the pressure at a depth of 40 feet.

SOLN: Another case of Direct Variation




(units are lb/ft3)




Use k = 5 lb/ft3 in the Direct Variation Equation to find the pressure at a depth of 40ft



Inverse VariationInverse Variation

INVERSE VARIATIONINVERSE VARIATION →

When a situation translates to an equation described by y = k/x, with k a constant, we Say that y varies INVERSELY as x. The equation y = k/x is called an equation of inverse variation

• Note that for k > 0, any equation of the form y = k/x indicates that as x increases, y decreases



Example Example Inverse Variation 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:

2030

k

x

ky

Solving for k

6002030 k

Thus the Equation of Variation

xy

600



Example Example Barn Building 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 Example Barn Building Barn Building cont.1 cont.1

1. Familarize. Think about the situation. What kind of variation applies? It seems reasonable that the greater number of people working on a job, the less time it will take. So LETLET:

• T ≡ the time to complete the job, in hours,

• N ≡ the number of people working

Then as N increases, T decreases and inverse variation applies




2. Translate: Since inverse variation applies use N

kT

3. Carry Out: Find the Constant of Inverse Proportionality

N

kT workers25

56k

hrs

Hrs Worker10082556 k




3. Carry Out: The Eqn of Variation

workers35

hrs•008worker1T

When N = 35. Find T

N

kT

hrshrsT 298.28

4. Chk: A check might be done by repeating the computations or by noting that (28.8)(35) and (56)(25) are both 1008.




5. STATE: if It takes 56 hours for 25 people to raise a barn, then it should take 35 people about 29 hours to build the same barn



To Solve Variation ProblemsTo Solve Variation Problems

1. Determine from the language of the problem whether direct or inverse variation applies.

2. Using an equation of the form y = kx for direct variation or y = k/x for inverse variation, substitute known values and solve for k.

3. Write the equation of variation and use it, as needed, to find unknown values.



Applications Tips ReDuxApplications Tips ReDux

The Most Important Part of Solving REAL WORLD (Applied Math) Problems

The Two Keys to the Translation• Use the LETLET Statement to ASSIGN

VARIABLES (Letters) to Unknown Quantities

• Analyze the RELATIONSHIP Among the Variables and Constraints (Constants)



Solving Variation ProblemsSolving Variation Problems

1. Write the equation with the constant of variation, k.

2. Substitute the given values of the variables into the equation in Step 1 to find the value of the constant k.

3. Rewrite the equation in Step 1 with the value k from Step 2

4. Use the equation from Step 3 to answer the question posed in the problem.



Other Variation RelationsOther Variation Relations

Some Additional Variation Eqns: y varies directly as the nth power of x

if there is some positive constant k such that

ny kx

n

ky

x

y varies inversely as the nth power of x if there is some positive constant k such that

y varies jointly as x and z if there is some positive constant k such that.

kxzy



Combined VariationCombined Variation

The Previous Variation Forms can be combined to create additional equations

z varies directly as x and INversely as y if there is some positive constant k such that

w varies jointly as x & y and inversely as z to the nth power if there is some positive constant k such that

y

xkz

nz

xykw



Example Example Luminance Luminance

The Luminance of a light (E) varies directly with the intensity (I) of the light source and inversely with the square distance (D) from the light. At a distance of 10 feet, a light meter reads 3 units for a 50-cd lamp. Find the Luminance of a 27-cd lamp at a distance of 9 feet.

This is a case of COMBINED Variation →



Example Example Luminance Luminance

Solve for the Variation Constant, k,Using the KNOWN values of I & D

2

2

503

106

IE k

Dk

k

Use the value of k, and

D = 9ft in the variation eqn to find E(9ft)

State: At 9ft the 27cd Lamp produces a Luminance of 2 cd/m2

2

6 27

92

E

E



Example Example Sphere Volume Sphere Volume

Suppose that you had forgotten the formula for the volume of a sphere, but were told that the volume V of a sphere varies directly as the cube of its radius r. In addition, you are given that V = 972π when r = 9in.

Find V when r = 6in SOLUTION: Recognize as Direct

Variation to a Power: V = kr3




Now use KNOWN data to solve for k

V kr3

972 k 9 3

972 k 729

k 972729

k 4

3

Now Substitute k = 4π/3 into the Eqn of Variation

V 4

3r3




Finally Substitute r = 6 and solve for V(6) as requested

Using π ≈ 3.14159 find the Volume for a 6 inch radius sphere, V(6) ≈ 904.78 in3

V 4

3 6 3

288 cubic inches



Example Example Newton’s Law Newton’s Law

Newton’s Law of Universal Gravitation says that every object in the universe attracts every other object with a force acting along the line of the centers of the two objects and that this attracting force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the two objects.




Write the Gravitation Law Symbolically

SOLUTION: Let m1 and m2 be the masses of the two objects and r be the distance between them; a Diagram:




Next LET:• F ≡ the gravitational force between the

objects

• G ≡ the Constant of Variation; a.k.a., the constant of proportionality

Thus Newton Gravitation Law in Symbolic form

F G m1m2

r2 .



Example Example Newton’s Law Newton’s Law The constant of proportionality G is called the

universal gravitational constant. It is termed a “universal constant” because it is thought to be the same at all places and all times and thus it universally characterizes the intrinsic strength of the gravitational force.

If the masses m1 and m2 are measured in kilograms, r is measured in meters, and the force F is measured in newtons, then the value of G:

2

311

skg

m 10676

.G




Next Estimate the value of g; the “acceleration due to gravity” near the surface of the Earth. Use these estimates:

• Radius of Earth RE = 6.38 x 106 meters

• Mass of the Earth ME = 5.98 x 1034 kg

SOLUTION: By Newton’s 1st Law Force = (mass)·(acceleration) →

F ma




Now the “Force of Gravity” at the earth’s surface is the result of the Acceleration of Gravity:

gmF

Equating the “Force of Gravity” and the Gravitation Force Equations:

Fgmr

mmGF

221




CarryOut

g 6.67 10 11 5.98 1024

6.38 106 2

g 9.8 m/sec2

mg G mM E

RE2

g G M E

RE2



Example Example Kinetic Energy Kinetic Energy

The kinetic energy of an object varies directly as the square of its velocity.

If an object with a velocity of 24 meters per second has a kinetic energy of 19,200 joules, what is the velocity of an object with a kinetic energy of 76,800 joules?

SOLUTION: This is case of Direct Variation to the Power of 2




Write the Equation of Variation

Next Solve for the Variation Constant, k, using the known data




To find k, use the fact that an object with a velocity of 24 m/s has a kinetic energy of 19.2 kJ

Thus k = 33.33 J/m2




Use k = 33.33 J/m2 to refine the Variation Equation

Next use the E(v) eqn to find v for E = 76.8 kJ2




The v for E = 76.8 kJ

Thus when E = 76.8 kJ the velocity is 48 m/s



WhiteBoard WorkWhiteBoard Work

Problems From §6.8 Exercise Set• 33, 38

KINETIC andPOTENTIALEnergyBalance



All Done for TodayAll Done for Today

Heat FlowsHot→Cold



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22



Graph Graph yy = | = |xx||

Make T-tablex y = |x |

-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6

x

y

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-5

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0

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-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls



x

y

-3

-2

-1

0

1

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-3 -2 -1 0 1 2 3 4 5

M55_§JBerland_Graphs_0806.xls -10

-9

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-3

-2

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0

1

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-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls

xy

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