9.6 Rational Equation word problems Work problems Motion

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9.6 Rational Equation word problems

• Work problems
• Motion problems

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Work problems

• Suppose Peter can paint a house in 8 hours and
  Danny can paint the same house in 11 hours.
  Assuming they work with maximum efficiency, how
  long would it take them to paint the house if
  they work together?
• The reasoning for this problem involves looking
  at the fraction of the job each person completes
  (hence rational equations), and we generally want
  these fractions to add up to 1 whole job.
• For example, in 1 hr Pete paints 1/8 of the house
• In 2 hrs he paints 2/8 of the house
• In 3 hrs he paints 3/8 of the house, and so on
• Similarly, in 1 hr Danny paints 1/11 of the house
• In 2 hrs he paints 2/11 of the house, and so on

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Painting example, continued

• Let X the of hours it will take Danny and
  Peter to complete the job together
• In x hrs, Danny completes x/11 of the job
• In x hrs, Peter completes x/8 of the job
• And together these fractions should add up to 1
  whole job
• That is, (x/8) (x/11) 1
• Now multiply by the LCM (88)
• 11x 8x 88
• 19x 88
• X 88/19 4.63 hours or 4 hours 37 min 48 sec
  P

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Example 6-3a
Mowing Lawns Tim and Ashley mow lawns together.
Tim working alone could complete the job in 4.5
hours, and Ashley could complete it alone in 3.7
hours. How long does it take to complete the job
when they work together?
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Example 6-3b
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Example 6-3c
Solve the equation.
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Example 6-3d
Answer It would take them about 2 hours working
together.
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Example 6-3e
Cleaning Libby and Nate clean together. Nate
working alone could complete the job in 3 hours,
and Libby could complete it alone in 5 hours. How
long does it take to complete the job when they
work together?
Answer about 2 hours
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Motion problems

• We represent motion at a constant speed using
  the formula
• d r t
• Where d distance
• r rate (speed mph, ft per
  sec..)
• t time
• Generally given two of these quantities you can
  determine the 3rd

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Usually 9.6 motion problems are more complicated

• You may have the motion of 2 (or sometimes more
  objects)
• Sometimes you will need to use formulas like
• Time 1 time 2 total time
• Or sometimes
• Distance 1 Distance 2 total distance
• Usually you will then replace the terms in the
  sum on the left side with an expression involving
  the other variables
• For example, since t d / r
• Time 1 time 2 total time
• Can become
• (distance 1 / rate 1) (distance 2 / rate 2)
  total time

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Example 6-4a
Swimming Janine swims for 5 hours in a stream
that has a current of 1 mile per hour. She leaves
her dock and swims upstream for 2 miles and then
back to her dock. What is her swimming speed in
still water?
Variables Let r be her speed in still water. Then
her speed with the current is r 1 and her speed
against the current is r 1.
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Example 6-4b
Solve the equation.
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Example 6-4c
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Example 6-4d
Use the Quadratic Formula to solve for r.
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Example 6-4e
Answer Since the speed must be positive, the
answer is about 1.5 miles per hour.
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Example 6-4f
Swimming Lynne swims for 1 hour in a stream that
has a current of 2 miles per hour. She leaves her
dock and swims upstream for 3 miles and then back
to her dock. What is her swimming speed in still
water?
Answer about 6.6 mph
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Other motion problems

• Sometimes you will have a motion problem with 2
  objects where one of the quantities for the 2
  objects is the same
• For example, distance 1 distance 2
• Then you can again substitute the other variables
  by using d r t
• So that distance 1 distance 2 becomes
• (rate 1) (time 1) (rate 2) (time 2)

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Train example

• The JT Express leaves the station 3.125 hours
  after a freight train leaves the same depot. The
  freight is travelling 25 mph slower than the JT
  Express. Find the rate of each train if the
  passsenger train overtakes the freight train in 5
  hours.

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Making a table

• Sometimes it helps where we write what we know
  about each object in a table
• distance rate time
• JT same x 5
• Freight same x 25 8.125
• Note that since the distance is the same for each
  object, distancejt distancefr

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Finishing

• Distancejt distancefr
• Ratejt timejt ratefr timefr
• X 5 (x 25) 8.125
• 5x 8.125x 203.125
• -3.125x -203.125
• X 65
• So JT EXPRESS travels at 65 mph, and the freight
  train travels at 40 mph
• Check 65 (5) 40 (8.125) both equal 325 miles

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HOMEWORK!

• In the text try pg. 509 10 and 36
• Also try the ODD problems on the worksheet.. Some
  are challenging
• You will have the even problems as part of your
  homework next time!!