4.6 -Modeling and Applications of Rational Functions
Recall that a rational expression is the ratio of two po lynomials. Rational expressions can be used to so lve
problems that involve comparing two quantities of the same unit of measure.
Example 1: Delilah is making her own sa lad dressing out of red vinegar and olive oil. It's a new recipe so
she has to determine the correct proportions. She mixes 10 teaspoons of vinegar and 16 teaspoons of olive
oil. After she stirs the mixture, she realize s it's not the consistency she wants, so she adds more olive oil.
a.
What is the ratio of red vinegar to olive oil
if she adds 6 teaspoons more of olive oil?
b. What is the ratio of red vinegar to olive oil
if she adds 10 teaspoons more of olive oil?
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c.
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Write an expression to represent the ratio of red vinegar to olive oil. Let x represent the number of
additional teaspoons of olive oil added to the recipe.
d. The recommended ratio of vinegar to olive is 1:7. Determine the amount of olive oil that she must
add to the mixture.
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Example 2: A door-to-door salesperson for a TV Cable Company offers cab le television for only $55.95 per
month. However, there is a one-time installation cost of $180.
a.
Determine the total cost of the cable for
the first two months.
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b. What is the average cost per month over
the first two months?
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c.
Write an equation to represent the total cost
of cable for x months.
d. Write an equation to represent the average
monthly cost of cable for x mqnths.
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e.
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A competitor offers a similar product for $65 per month and no installation charges. Who is
offering the better deal? Explain .
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Unit4
1-! Secondary 3
A work problem is a type of problem that involves the rates of several workers and the time it takes to
complete a job. For example, the rate at wh ich two painters work and the total time it takes them to paint
a house while working together is an example of a work problem.
Example 3: Maureen is a community volunteer. She volunteers by watering the large vegetable garden in
her neighborhood. Sometimes, Maureen's friend Sandra also volunteers.
It takes Ma ureen 90 minutes to water the garden. When Maureen and Sandra are working together, they
can complete the job in 40 minutes.
a. Complete the table. Let x represent the time it takes Sandra to water the garden if she works
alone.
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Portion of the
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b.
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Maureen
Entiro Job, or
1 Garden
Time Spont
Wabtnng
Garden Watered
40
40
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Write and solve an equation to determine the total time it would take Sandra to water the garden
if she were working alone.
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A mixture problem is a type of problem that involves the combination of two or more liquids and the
concentrations of those liquids.
Example 4: Manuel is taking a college chemistry course, and some of his time is spent in the chemistry lab.
He is conducting an experiment for which he needs a 2%sa lt solution. However, all he can find in the lab is
120 milliliters (mL) of 10% salt solution.
a.
How many milliliters of salt and how many milliliters of water are in 120mL of 10% sa lt solution?
b.
What would the concentration of the salt solution be if Manuel added 80mL of water?
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c.
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Write and solve an equation to calculate the amount of water Manuel needs to add to the 120mL
of 10% salt solution to make a 2% sa lt solution. Let x represent the amount of water Manuel needs
to add.
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Example 5: Keisha is working on a chemistry experiment. Suppose that the 20mL of a 20% sulfuric acid is
mixed with lOmL of the 5% sulfuric acid so lution.
a.
What is the concentration of the resulting solution?
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b.
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Write and solve an equation to calculate the amount of 5% sulfuric acid solution Keisha added if
the resulting solution is a 12% sulfuric acid solution. Let x represent the amount of 5% sulfuric acid
that she is adding.
Unit4
H Secondary 3
A distance problem is a type of problem that involves distance, rate and time.
D =RT
Example 6: A river barge travels 140 miles from a loading dock to a warehouse to deliver supplies.
Then, the barge returns to the loading dock. The barge travels with the current to the warehouse and
against the current from the warehouse. The barge's total travel times is 20 hours, and it travels in still
water at an average speed of 15 miles per hour.
a.
Use the given information to complete the table. Let x represent the average speed of the current.
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b.
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Distance Traveled
Average Speed
Time Traveled
Mtles
Miles
Hours
Hours
\%
15 +X
Against the
Current
140
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Round Trip
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With the Current
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You are given that the barge's total travel time is 20 hours. Write an algebraic expression, in terms
of the number of hours the barge travels with the current and the number of hours it travels
against the current, which is equivalent to 20 hours.
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c.
Write and solve an equation to calculate the average speed of the current.
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A cost problem is a type of problem that involves the cost of ownership of an item over time.
Example 7: Melinda has decided that it is time to replace her old refrigerator. She purchases a new
refrigerator that uses less electricity than the old one. Melinda purchases her new refrigerator for
$2000. The refrigerator costs $46 per year to operate.
a.
Write an expression to represent Melinda's average annual cost of owning the new refrigerator
for x years.
b.
When Melinda's average annual cost of owning the refrigerator is less than $400, she plans to
shop for a new television. When can Melinda shop for a new television?
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