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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.
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.

Tiffaniqua and Mark flying a drone
External credits: macrovector

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.
b What is the speed of the wind if the total time of the trip is minutes?
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.
Which option will give a better combined mpg?

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.

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.
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.
miles mpg
SUV
Hybrid
By substituting these values into the formula, the combined mpg is calculated.
Simplify right-hand side
In this option, the combined mpg is

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
Substitute these expressions into the formula.
Simplify right-hand side
In this option, the combined mpg is

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.

A map that shows lake, forrest, some buildings, and road construction
This resulted in the return trip taking hours longer. How many hours did it take them to get home from the store?

Hint

The distance traveled is equal to the product of the speed and the time.

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
Since the distance traveled is equal to the product of the speed and the time, a system of equations can be written.
By manipulating the equations in the system, rational equations can be formed.
Next, since is isolated in Equation (I), its equivalent expression can be substituted into Equation (II).
Now, the second equation is written only in terms of Since only the time is required, this equation will be solved for the time
Simplify
This equation is now a quadratic equation in terms of It can be solved by factoring.
Factor
Solve using the Zero Product Property
Since time cannot be negative, the solution is disregarded. Therefore, is the only solution in this context. Recall that represents the time to get home from the store. This means that it took Tiffaniqua hours to get home from the store.
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.

ChemMixing.svg

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.
b After how many minutes will the concentration be

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.
The concentration is the ratio of grams of resin to milliliters of water.
b To find after how many minutes the concentration will be substitute for into the equation found in Part A. Then, solve it for
Solve for
After minutes, the concentration in the flask will be
Using constant rates of change, Tiffaniqua and Mark have successfully made the special mixture for the filament that they hope will become an awesome printed drone — a drone that may have more purpose than what they expect.
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.

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?

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.
Simplify
Rewrite
To solve this quadratic equation, the Quadratic Formula can be used.
Solve using the quadratic formula
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.

Evaluate left-hand side
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.
Rewrite
The answer is hours and minutes.
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.

remote controller and drone
External credits: macrovector

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.
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.

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.
A combined variation equation can then be written as follows.
Here is the constant of variation and cannot be To find the value of review the given information.
Substitute these values into the equation and solve for
Solve for

Write in scientific notation

The constant of variation is
b Consider the equation written in Part A.
When is less than or equal to milliwatts, the connection is lost. To keep the drone under control, the drone must be within a certain distance. This distance can be found by solving the following equation.
The same values from Part A can be used for and as they are the same for this case as well.
Solve for

The negative solution is ignored. The drone can be operated within a distance of about meters.
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.

Tiffaniqua and Mark flying a drone
External credits: macrovector

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.
b What is the speed of the wind if the total time of the trip is minutes?

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.
Since is the speed of the wind in kilometers per hour, will be the speed of the drone while going upwind and will be the speed of the drone while going downwind. The information about the distance, speed, and time can be organized in a table.
Upwind Downwind
Distance (km)
Speed (km/h)
Time (h)
Therefore, the flight time of the drone is the sum of the times.
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
Now substitute this value for and solve the equation for