[email protected] mth55_lec-35_sec_6-7_rational_equation_applications.ppt 1 bruce mayer, pe...
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[email protected] • MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§6.7 Rational§6.7 RationalEqn AppsEqn Apps
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Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §6.6 → Rational Equations
Any QUESTIONS About HomeWork• §6.6 → HW-22
6.6 MTH 55
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Bruce Mayer, PE Chabot College Mathematics
§6.7 Rational Equation Applications§6.7 Rational Equation Applications
Problems Involving Work
Problems Involving Motion
Problems Involving Proportions
Problems involving Average Cost
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Bruce Mayer, PE Chabot College Mathematics
Solve a Formula for a VariableSolve a Formula for a Variable
Formulas occur frequently as mathematical models. Many formulas contain rational expressions, and to solve such formulas for a specified letter, we proceed as when solving rational equations.
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Bruce Mayer, PE Chabot College Mathematics
Solve Rational Eqn for a VariableSolve Rational Eqn for a Variable1. Determine the DESIRED letter (many times
formulas contain multiple variables)2. Multiply on both sides to clear fractions or
decimals, if that is needed. 3. Multiply if necessary to remove parentheses. 4. Get all terms with the letter to be solved for on one
side of the equation and all other terms on the other side, using the addition principle.
5. Factor out the unknown.6. Solve for the letter in question, using the
multiplication principle.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Solve for Letter Solve for Letter
Solve this formula for y: .aT
RT ay
SOLN:aT
RT ay
T ay T aaT
T ayyR
R T ay aT
RT Ray aT
Ray aT RT
aT RTy
Ra
Multiplying both sides by the LCD
Simplifying
Dividing both sides by Ra
Multiplying
Subtracting RT
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Bruce Mayer, PE Chabot College Mathematics
Example Example Fluid Mechanics Fluid Mechanics
In a hydraulic system, a fluid is confined to two connecting chambers. The pressure in each chamber is the same and is given by finding the force exerted (F) divided by the surface area (A). Therefore, we know
Solve this Eqn for A2
1 2
1 2
.F F
A A
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Bruce Mayer, PE Chabot College Mathematics
Example Example Fluid Mechanics Fluid Mechanics
SOLUTION:
This formula can be used to calculate A2 whenever A1, F2, and F1 are known
1 2
11
22 1 2A
AA A A
F F
A
2 1 1 2A F A F
1 22
1
A FA
F
Multiplying both sides by the LCD
Dividing both sides by F1
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Bruce Mayer, PE Chabot College Mathematics
Problems Involving WorkProblems Involving Work
Rondae and Marrisa work during the summer painting houses. • Rondae can paint an average size
house in 12 days
• Marrisa requires 8 days to do the same painting job.
How long would it take them, working together, to paint an average size house?
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Bruce Mayer, PE Chabot College Mathematics
House Painting cont.House Painting cont.
1. Familiarize. We familiarize ourselves with the problem by exploring two common, but incorrect, approaches.
a) One common, incorrect, approach is to add the two times. → 12 + 8 = 20
b) Another incorrect approach is to assume that Rondae and Marrisa each do half the painting.
– Rondae does ½ in 12 days = 6 days– Marrisa does ½ in 8 days = 4 days – 6 days + 4 days = 10 days.
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Bruce Mayer, PE Chabot College Mathematics
House Painting cont.House Painting cont. A correct approach is to consider how much
of the painting job is finished in ONE day; i.e., consider the work RATE
It takes Rondae 12 days to finish painting a house, so his rate is 1/12 of the job per day.
It takes Marrisa 8 days to do the painting alone, so her rate is 1/8 of the job per day.
Working together, they can complete 1/8 + 1/12, or 5/24 of the job in one day.
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Bruce Mayer, PE Chabot College Mathematics
House Painting cont.House Painting cont.
Note That given a TIME-Rate
[Amount] = [Rate]•[TimeQuantity]
t/8t1/8Marrisa
t/12t1/12Rondae
Amount Completed
TimeRate of Work
Painter
Form a table to help organize the info:
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Bruce Mayer, PE Chabot College Mathematics
House Painting cont.House Painting cont.
2. Translate. The time that we want is some number t for which
1 11
12 8t t Portion of work done by
Marrisa in t daysPortion of work done by Rondae in t days
Or 1 o1 1
r 5
1 1.
2 8 24t t
Portion of work done together in t days
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Bruce Mayer, PE Chabot College Mathematics
House Painting cont.House Painting cont.
3. Carry Out. We can choose any one of the above equations to solve:
51
24t
51
24
24 24
5 5t
24 4, or 4 days
5 5t
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Bruce Mayer, PE Chabot College Mathematics
House Painting cont.House Painting cont.
4. Check. Test t = 24/5 days
24 241
5 5
1 2 3 51
12 8 5 5 5
5. State. Together, it will take Rondae & Marrisa 4 & 4/5 days to complete painting a house.
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Bruce Mayer, PE Chabot College Mathematics
The WORK PrincipleThe WORK Principle
Suppose that A requires a units of time to complete a task and B requires b units of time to complete the same task.
Then A works at a rate of 1/a tasks per unit of time.
B works at a rate of 1/b tasks per unit of time,
Then A and B together work at a totalrate of [1/a + 1/b] per unit of time.
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Bruce Mayer, PE Chabot College Mathematics
The WORK PrincipleThe WORK Principle
If A and B, working together, require t units of time to complete a task, then their combined rate is 1/t and the following equations hold:
111
tb
ta
111
bat
1b
t
a
t
tba
111
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Bruce Mayer, PE Chabot College Mathematics
Problems Involving MotionProblems Involving Motion
Because of a tail wind, a jet is able to fly 20 mph faster than another jet that is flying into the wind. In the same time that it takes the first jet to travel 90 miles the second jet travels 80 miles. How fast is each jet traveling?
r r+20
HEAD Wind TAIL Wind
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Bruce Mayer, PE Chabot College Mathematics
HEADwind vs. TAILwindHEADwind vs. TAILwind
1. Familiarize. We try a guess. If the fast jet is traveling 300 mph because of a tail wind the slow jet plane would be traveling 300−20 or 280 mph.
• At 300 mph the fast jet would have a 90 mile travel-time of 90/300, or 3/10 hr.
• At 280 mph, the other jet would have a travel-time of 80/280 = 2/7 hr.
Now both planes spend the same amount of time traveling, So the guess is INcorrect.
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Bruce Mayer, PE Chabot College Mathematics
HEADwind vs. TAILwindHEADwind vs. TAILwind
2. Translate. Fill in the blanks using
[TimeQuantity]=[Distance]/[Rate]
AirCraft
Distance(miles)
Speed or Rate(miles per hour)
Time(hours)
Jet 1 80 r
Jet 2 90 r + 20
r r+20
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Bruce Mayer, PE Chabot College Mathematics
HEADwind vs. TAILwindHEADwind vs. TAILwind
Set up a RATE Table
[Distance]/[Rate] = [TimeQuantity]
AirCraft
Distance(miles)
Speed(miles per hour)
Time(hours)
Jet 1 80 r 80/r
Jet 2 90 r + 20 90/(r + 20)
The Times MUST be the SAME
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Bruce Mayer, PE Chabot College Mathematics
HEADwind vs. TAILwindHEADwind vs. TAILwind
Since the times must be the same for both planes, we have the equation
3. Carry Out. To solve the equation, we first Clear-Fractions multiplying both sides by the LCD of r(r+20)
( 20) (80 9
02 )
20
0r r r r
r r
20
9080
rt
r
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Bruce Mayer, PE Chabot College Mathematics
HEADwind vs. TAILwindHEADwind vs. TAILwind
Complete the “Carry Out”
80( 20) 90r r
80 1600 90r r 1600 10r
160 r
Simplified by Clearing Fractions
Using the distributive law
Subtracting 80r from both sides
Dividing both sides by 10
Now we have a possible solution. The speed of the slow jet is 160 mph and the speed of the fast jet is 180 mph
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Bruce Mayer, PE Chabot College Mathematics
HEADwind vs. TAILwindHEADwind vs. TAILwind
4. Check. ReRead the problem to confirm that we were able to find the speeds. At 160 mph the jet would cover 80 miles in ½ hour and at 180 mph the other jet would cover 90 miles in ½ hour. Since the times are the same, the speeds Chk
5. State. One jet is traveling at 160 mph and the second jet is traveling at 180 mph
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Bruce Mayer, PE Chabot College Mathematics
Formulas in EconomicsFormulas in Economics
Linear Production Cost Function
C x variable cost fixed costs ax b
• Where– b is the fixed cost in $
– a is the variable cost of producing each unit in $/unit (also called the marginal cost)
AverageCost ($/unit) C x C x
x
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Bruce Mayer, PE Chabot College Mathematics
Formulas in EconomicsFormulas in Economics
Price-Demand Function: Suppose x units can be sold (demanded) at a price of p dollars per units.
• Where– m & n are SLOPE Constants in $/unit & unit/$
– d & k are INTERCEPT Constants in $ & units
p x mx d
or
x p np k
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Bruce Mayer, PE Chabot College Mathematics
Formulas in Economics
Revenue Function
Revenue = (Price per unit)·(No. units sold)
xdmxxpxR
Profit FunctionProfit = (Total Revenue) – (Total Cost)
P x R x C x
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Bruce Mayer, PE Chabot College Mathematics
Example Average Cost
Metro Entertainment Co. spent $100,000 in production costs for its off-Broadway play Pride & Prejudice. Once running, each performance costs $1000
a) Write the Cost Function for conducting z performances
b) Write the Average Cost Function for the z performances
c) How many performances, n, result in an average cost of $1400 per show
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Bruce Mayer, PE Chabot College Mathematics
Example Example Average Cost Average Cost
SOLUTION a) Total Cost is the sum of the Fixed Cost and the Variable Cost
zzCshow
k1k100
$$
SOLUTION b) The Average Cost Fcn
z
z
z
zCzC show
k1k100
$$
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Bruce Mayer, PE Chabot College Mathematics
Example Example Average Cost Average Cost
SOLUTION c) In this case for “n” Shows
n
n
n
nnC
k1$k100$showk1$
k100$k4.1$
nn k1$k100k4.1$ k100k4.0 n
k4.0
k100k4.0 n250n
Thus 250 shows are needed to realize a per-show cost of $1400
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Bruce Mayer, PE Chabot College Mathematics
Problems Involving ProportionsProblems Involving Proportions Recall that a RATIO of two quantities is
their QUOTIENT.• For example, 45% is the ratio of 45 to 100,
or 45/100.
A proportion is an equation stating that two ratios are EQUAL:
An equality of ratios, An equality of ratios, AA//BB = = CC//DD, is , is called a PROPORTION. The numbers called a PROPORTION. The numbers
within a proportion are said to be within a proportion are said to be proportionAL to each otherproportionAL to each other
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Bruce Mayer, PE Chabot College Mathematics
Example Example Triangle Triangle ProportionsProportions Triangles ABC and XYZ are “similar”
A
B
C X
Y
Z
a = 7
b
x = 8
y = 12
Now Solve for b if
x = 8, y = 12 and a = 7
• Note that “Similar” Triangles are “In Proportion” to Each other
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Bruce Mayer, PE Chabot College Mathematics
Example Example Similar Triangles Similar Triangles
Set Up TheProportions
A
B
C
X
Y
Z
a = 7
b
x = 8
y = 12
7
12 8
b
712
8b
84 or 10.5
8b
[b is to 12]as
[7 is to 8]
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Bruce Mayer, PE Chabot College Mathematics
Example Example Similar Triangles Similar Triangles
AlternativeProportions
A
B
C
X
Y
Z
a = 7
b
x = 8
y = 12
84 or 10.5
8b
[b is to 7]as
[12 is to 8]
8
12
7
b
8
127b
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Bruce Mayer, PE Chabot College Mathematics
Example Example Quantity Proportions Quantity Proportions
A sample of 186 hard drives contained 4 defective drives. How many defective drives would be expected in a group of 1302 HDDs?
Form a proportion in which the ratio of defective hard drives is expressed in 2 ways.
4
186 1302
x
defective drives
total drives
defective drives
total drives
186 5208x
28x
Expect to find 28 defective HDDs
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Bruce Mayer, PE Chabot College Mathematics
Whale ProportionalityWhale Proportionality
To determine the number of humpback whales in a pod, a marine biologist, using tail markings, identifies 35 members of the pod.
Several weeks later, 50 whales from the SAME pod are randomly sighted. Of the 50 sighted, 18 are from the 35 originally identified. Estimate the number of whales in the pod.
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Bruce Mayer, PE Chabot College Mathematics
Tagged Whale ProportionsTagged Whale Proportions
1. Familarize. We need to reread the problem to look for numbers that could be used to approximate a percentage of the of the pod sighted.
Since 18 of the 35 whales that were later sighted were among those originally identified, the ratio 18/50 estimates the percentage of the pod originally identified.
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Bruce Mayer, PE Chabot College Mathematics
HumpBack WhalesHumpBack Whales
2. Translate: Stating the Proportion
35 18
50w
Marked whales sighted later
Total Whales sighted later
Whales originally Marked
Total Whales in pod
3. CarryOut
35 18
550 50
0w
ww
50 35 or 97.22
18w
50 35 18 w
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Bruce Mayer, PE Chabot College Mathematics
More On WhalesMore On Whales
4. Check. The check is left to the student.
5. State. There are about 97 whales in the Pod
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Bruce Mayer, PE Chabot College Mathematics
One More WhaleOne More Whale
Another way to summarize the RANDOM-Tagging and RANDOM-Sighting Relation:
[35 is to w]as
[18 is to 50]
Thus theProportionality:
Solve for w18
50
35
w
2.9718
1750
18
5035
18
50
3535
w
w
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Bruce Mayer, PE Chabot College Mathematics
Example Example Vespa Scooters Vespa Scooters
Juan’s new scooter goes 4 mph faster than Josh does on his scooter. In the same time that it takes Juan to travel 54 miles, Josh travels 48 miles.
Find the speed of each scooter.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Vespa Scooters Vespa Scooters Familiarize. Let’s guess that Juan is
going 20 mph. Josh would then be traveling 20 – 4, or 16 mph.
At 16 mph, he would travel 48 miles in 3 hr. Going 20 mph, Juan would cover 54 mi in 54/20 = 2.7 hr. Since 3 2.7, our guess was wrong, but we can see that if r = the rate, in miles per hour, of Juan’s scooter, then the rate of Josh’s scooter = r – 4.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Vespa Scooters Vespa Scooters
LET: • r ≡ Speed of Juan’s Scooter
• t ≡ The Travel Time for Both Scooters
Tabulate the data for clarity
Distance Speed Time
Juan’s Scooter
Josh’s Scooter
54
48
r
4r
t
t
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Bruce Mayer, PE Chabot College Mathematics
Example Example Vespa Scooters Vespa Scooters
Translate. By looking at how we checked our guess, we see that in the Time column of the table, the t’s can be replaced, using the formula
Time = Distance/SpeedDistance Speed Time
Juan’s Scooter
Josh’s Scooter
54
48
r
4r
54 / r
48 /( 4)r
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Bruce Mayer, PE Chabot College Mathematics
Example Example Vespa Scooters Vespa Scooters
Since the Times are the SAME, then equate the two Time entries in the table as:
54 48.
4r r
CarryOut
54 48
4r r
54
(48
) 4)4
4 (r r rr
rr
54 216 48r r
216 6r 36 .r
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Bruce Mayer, PE Chabot College Mathematics
Example Example Vespa Scooters Vespa Scooters
Check: If our answer checks, Juan’s scooter is going 36 mph and Josh’s scooter is going 36 − 4 = 32 mph. Traveling 54 miles at 36 mph, Juan is riding for 54/36 or 1.5 hours. Traveling 48 miles at 32 mph, Josh is riding for 48/32 or 1.5 hours. The answer checks since the two times are the same.
State: Juan’s speed is 36 mph, and Josh’s speed is 32 mph
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §6.7 Exercise Set• 16 (ppt), 34, 44
Mass Flow Rate for aDivergingNozzle
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Bruce Mayer, PE Chabot College Mathematics
P6.7-16P6.7-16
Given Avg CostFunction Graph:
Find ProductionQuatity for Avg Cost of $425/Chair
SOLUTION: CastRight & Down
20k
ANS → 20k Chairs/mon
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Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
HumanProportions:HeadLength
BaseLine
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Bruce Mayer, PE Chabot College Mathematics
Example Example Similar Triangles Similar Triangles
SOLUTION Examine the drawing, write a proportion, and then solve.
A
B
C X
Y
Z
a = 7
b
x = 8
y = 12
Note that side a is always opposite angle A, side x is always opposite angle X, and so on.
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
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Bruce Mayer, PE Chabot College Mathematics
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
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
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Bruce Mayer, PE Chabot College Mathematics
x
y
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls -10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
xy