Transcript 6_5x

Problems Involving Work
Example: Juan and Rebecca work during the summer painting houses. Juan
can paint an average size house in 12 days, while Rebecca requires 8 days to do
the same job. How long would it take them, working together, to paint an
average size house?
1. Introduction. A correct approach is to consider how much of the painting
job is finished in one day. It takes Juan 12 days to finish painting a house, so
his rate is 1/12 of the job per day. It takes Rebecca 8 days to do the painting
alone, so her rate is 1/8 of the job per day.
Let t = The time it takes both to complete the job
together.
Rate of
Work
Time
Amount
Completed
Juan
1/12
t
t/12
Rebecca
1/8
t
t/8
continued
2. Body. The time that we want is some number t for which
t
Portion of work done by
Juan in t days
12

t
1
Portion of work done by
Rebecca in t days
8
t
 t
24  
   1  24
 12 8 
2 t  3t  2 4
5t  2 4
t
24
5
3. Conclusion. Together, it will take Juan and Rebecca 4 4/5 days to complete
painting a house.
Modeling Work Problems
If
a = the time needed for A to complete the work alone,
b = the time needed for B to complete the work alone, and
t = the time needed for A and B to complete the work together,
then t  t  1 .
a
b
The following are equivalent equations that can also be used:
1
a
t 
1
b
t  1
and
1
a

1
b

1
t
.
Problems Involving Motion
Example: 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?
1. Introduction.
Let x = speed of slower jet (mph)
x + 20 = speed of faster jet (mph)
x + 20
x
Distance =
Distance (in
miles)
Rate

Time
Speed (in miles Time
per hour)
(in hours)
Jet 1
80
x
80/x
Jet 2
90
x + 20
90/(x + 20)
Fill in the blank column in the table.
Tim e 
D istance
R ate
or t 
d
r
2. Body Since the times must be the same for both planes, we
have the equation
80 x  1600  90 x
80
x

90
1600  10 x
x  20
80( x  20)  90 x
Cross Multiply
x  160
3. Conclusion. One jet is traveling at 160 mph and the second jet is traveling at
180 mph.