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Rational functions can be effective tools for modeling real-life situations. Some applications of rational functions involve speed, rate of work, and mixing problems. Rational functions even have applications in medicine and economics. This lesson will investigate the use of rational functions in various situations.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Finding the Speed of the Wind
Tiffaniqua and Mark designed and printed a drone. A neighbor hears about their feat and needs their help — she lost her dog Quipu! Before taking flight to find Quipo, first, they need to determine the flying conditions. The drone can travel at a maximum speed of kilometers per hour. On a windy day, it travels kilometers against the wind and then returns to the starting location.
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Suppose that the drone constantly flies at its maximum rate throughout the search.
a Write a rational function for the time of the flight where is the speed of the wind.
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b What is the speed of the wind if the total time of the trip is minutes?
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Example
Combined Miles per Gallon
Tiffaniqua wants to put some of her math skils to use to help her parents. Her mom drives an SUV that travels miles per gallon (mpg) and her father has a hybrid that travels They travel the same distance every month. Tiffaniqua realizes that she can find a way to improve the combined miles per gallon. She is considering two options.
| Option I | Tune the SUV to increase its mileage by mpg and keep the hybrid as it is. |
|---|---|
| Option II | Buy a new hybrid that can travel mpg and keep the SUV as it is. |
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Hint
The combined mpg is equal to the total miles divided by the total gallons. This can algebraically be written as
Solution
The combined mpg is equal to the ratio of the total miles to the the total gallons.
Let be the number of miles Tiffaniqua's mother drives in a month. Since her parents travel the same distance every month, also represents the number of miles Tiffaniqua's father drives in a month. Now an expression for each option can be written.
The number of gallons consumed by each car can be expressed in terms of which is the number of miles that the cars travel in a month. The number of gallons can be found by dividing miles by miles per gallons.
By substituting these values into the formula, the combined mpg is calculated.
In this option, the combined mpg is
Substitute these expressions into the formula.
In this option, the combined mpg is
Option I: Tune the SUV, Keep Hybrid as It Is
The SUV is tuned and its mileage is increased by mpg. The hybrid, on the other hand, is kept as it is.| miles | mpg | ||
|---|---|---|---|
| SUV | |||
| Hybrid |
SubstituteExpressions
Substitute expressions
Option II: Buy a New Hybrid, Keep SUV as It Is
In this option, it is suggested to buy a new hybrid with a mpg of and keep the SUV. As in the procedure followed in the previous option, the information at hand can be organized in a table.
| miles | mpg | ||
|---|---|---|---|
| SUV | |||
| Hybrid |
Conclusion
If Option I is chosen, the combined mpg will be about If Option II is chosen, it will be Therefore, the first option gives a better combined mpg.
Example
Adjusting Estimated Time of Arrival
Tiffaniqua's parents want to thank her for solving their car problem and drove to a store miles away in the next city over to buy a new printer that she has dreamed of for years. On the way back, there was a road closure due to a landslide. This caused them to drive mph slower.
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Solution
Only the distance to the store is known. The time it takes to go to the store and the speed are unknown. Then, let and be the time elapsed and the speed, respectively.
| Speed | Time | Distance | |
|---|---|---|---|
| To the store |
On the way home, Tiffaniqua's mother drove slower and the trip took hours longer.
| Speed | Time | Distance | |
|---|---|---|---|
| To the store | |||
| From the store |
▼
Simplify
SubFrac
Subtract fractions
Distr
Distribute
Distr
Distribute
CommutativePropAdd
Commutative Property of Addition
FactorOut
Factor out
▼
Solve using the Zero Product Property
Example
Concentration of a Mixture
Tiffaniqua and Mark are excited to print parts to make a drone using Tiffaniqua's new printer. They want to make a special mixture for the raw material, called the filament. Tiffaniqua filled a flask with milliliters of water. She then adds grams of water soluble polyester resin.
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Mark then adds more water at a rate of milliliters per minute and simultaneously adds more resin at a rate of grams per minute.
a Write a function for the concentration of resin in the flask after minutes.
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b After how many minutes will the concentration be
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Hint
a How can the amount of water after minutes be represented? How can the amount of resin after minutes be represented?
b Substitute the given value into the function found in Part A.
Solution
a Since water increases at a rate of milliliters per minute and resin at grams per minute, these are constant rates of change. Let be the amount of water and the amount of resin in the flask after minutes. Considering that the flask initially has milliliters of water and grams of resin, two functions can be written.
b To find after how many minutes the concentration will be substitute for into the equation found in Part A. Then, solve it for
Example
Finding the Printing Time of the New D Printer
Tiffaniqua and Mark realize they can use their filament with Mark's old printer in addition to the new printer. There is one major difference between the printers, however, it would take the old printer minutes longer than the new printer to create the same parts.
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Working together — some parts are printed with the new printer and some with the old printer — they can complete the task in hours and minutes. How long would it take the new printer to create the parts alone? {"type":"choice","form":{"alts":["<span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=85bd619d843e098eabd1064f14c0dcb6&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.338ex;height: 2.176ex; width: 1.162ex;\" ><\/span><\/span> hours and <span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=f721623ebae324dfc916f6a9968c8761&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.338ex;height: 2.176ex; width: 1.162ex;\" ><\/span><\/span> minutes","<span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=85bd619d843e098eabd1064f14c0dcb6&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.338ex;height: 2.176ex; width: 1.162ex;\" ><\/span><\/span> hours and <span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=fcf03b8f5f7a522e3dec1756f786f87c&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.338ex;height: 2.176ex; width: 2.325ex;\" ><\/span><\/span> minutes","<span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=600f06c69c20e950d635ba46914f298b&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.338ex;height: 2.176ex; width: 1.162ex;\" ><\/span><\/span> hours and <span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=fcf03b8f5f7a522e3dec1756f786f87c&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.338ex;height: 2.176ex; width: 2.325ex;\" ><\/span><\/span> minutes","<span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=f721623ebae324dfc916f6a9968c8761&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.338ex;height: 2.176ex; width: 1.162ex;\" ><\/span><\/span> hours and <span><span class=\"mwe-math-fallback-image-inline\" style=\"background-image: url('\/mediawiki\/index.php?title=Special:MathShowImage&hash=f721623ebae324dfc916f6a9968c8761&mode=mathml'); background-repeat: no-repeat; background-size: 100% 100%; vertical-align: -0.338ex;height: 2.176ex; width: 1.162ex;\" ><\/span><\/span> minutes"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
Hint
If is the printing time of the new printer, then is the printing time of the old printer.
Solution
Let be the time in minutes it takes the new printer to print the parts Tiffaniqua and Mark need. Then, will be the printing time for the old printer. In this situation, the work is to print the drone parts, which can be considered as Each machine's rate of work is then over the printing time.
When both printers work at the same time, the job is completed in hours and minutes. Since is defined in minutes, this time should also be expressed in minutes.
The combined rate of the printers is over the combined printing time Therefore, the combined rate is This rate should be equal to the sum of the individual rates of the printers.
The above rational equation can be solved for To eliminate the denominators, the equation can be multiplied by the least common denominator, which is the product of all the denominators.
To solve this quadratic equation, the Quadratic Formula can be used.
Next, the solutions can be individualized by using the positive and negative signs.
Since time cannot be negative, is ignored. The solution needs to be checked.
Since a true statement was obtained, is a solution to the equation. The new printer prints the parts needed in about minutes. The final step is to write this in hours.
The answer is hours and minutes.
▼
Simplify
▼
Rewrite
▼
Solve using the quadratic formula
\(0.004994 \dots \approx 0.005\ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
Example
How Far Can the Drone Go?
Drones communicate using radio waves on specific radio frequencies. The drone operates at a frequency of gigahertz (GHz). How far a drone can travel depends on a number of factors such as the power of the signal transmitted by the controller.
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Let be the power of the radio signal transmitted by the controller, the power of the radio signal received by the drone, the wavelength, and the distance between the drone and its controller. It is known that varies directly with and the square of and inversely with the square of
a The controller transmits milliwatts (mW) of power. When the distance between the drone and the controller is meters, the power received by the drone is milliwatts. The wavelength of a signal is about meters. Find the constant of variation.
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b If the power received is less than or equal to milliwatts, data exchange is not possible with the drone. How far can the drone go? Round the answer to the nearest integer.
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Hint
a Use the equation of the combined variation where is the constant of variation.
b Substitute the given value into the equation.
Solution
a It is given that varies directly with and and inversely with Start by labeling the quantities.
SubstituteValues
Substitute values
▼
Solve for
CalcPow
Calculate power
Multiply
Multiply
CalcQuot
Calculate quotient
Write in scientific notation
WriteProdFrac
Write as a product of fractions
CalcQuot
Calculate quotient
CalcPow
Calculate power
Multiply
Multiply
RearrangeEqn
Rearrange equation
b Consider the equation written in Part A.
Closure
Calculating the Speed of the Wind
From solving her parents' car issues, to successfully printing a drone, Tiffaniqua and Mark are now trying to locate a missing dog, Quipu! The owner tells them that Quipu was last seen running along the road that leads to the next city where Tiffaniqua bought the printer. They desperately want to find Quipo quickly. The drone can fly at a max speed of kilometers per hour.
Error loading image.
Before setting out to find Quipo, they test flew the drone kilometers against the wind before needing to bring it back to the starting point. Suppose that the drone flew at its maximum rate of km/h throughout the trip.
a Write a rational function for the time of the flight where is the speed of the wind.
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b What is the speed of the wind if the total time of the trip is minutes?
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Hint
a Write two rational expressions for the times it takes to go upwind and downwind.
b Convert the given time into hours.
Solution
a Recall that the distance is the product of the rate and the time.
| Upwind | Downwind | |
|---|---|---|
| Distance (km) | ||
| Speed (km/h) | ||
| Time (h) |
b In the previous part, the unit of is hours. For this reason, the given time must be converted into hours. To do so, is multiplied by the conversion factor
▼
Solve for
Distr
Distribute
CancelCommonFac
Cancel out common factors
SimpQuotProd
Simplify quotient and product
Distr
Distribute
AddSubTerms
Add and subtract terms
CalcRoot
Calculate root
▼
Evaluate right-hand side
\(\dfrac{5}{12} = \dfrac{5}{12} \ {\color{#009600}{\bm{\Large{\checkmark }}}}\)
They begin flying the drone going downwind, making the drone and speed of the search faster. Over a few hours, they have no luck or clue about Quipu. Tiffaniqua's got it — she remembers the landslide along the road Quipu was last seen. Since the drone can only cover about meters from the controller, they start closer to the landslide.
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Maybe Quipu went there and became stuck among some huge rocks? The drone flies over hard-to-reach places, searching behind colossal rock after colossal rock. The drone is thrown off route by a gust of wind and hits one of the rocks, but thanks to the great filament, it continues searching, undamaged. "Quipu!" cries out Tiffaniqua.
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