4.07 Modeling with rational functions

Examine the equation, graph and table that model this context. Are they all accurate models of the situation? If so, explain how the model matches the context. If not, identify any problems with the model, and revise the model to address the problems.

It is possible to consider the representations of the context in any order. For this context, we can start by reviewing the table of values to confirm that the time, water, and salt align to the given scenario. Then, we can work with the other representations to determine their accuracy.

\dfrac{120 \text{ cups}}{1 \text{ hour}} \times \dfrac{1 \text{ hour}}{60 \text{ minutes}}= \dfrac{2 \text{ cups}}{1 \text{ minute}}

The rate at which the salt is being poured would appear in the table as 10 cups every 5 minutes, which is accurate. Finally, we should align the units of the water and salt measurements in the table so that they are more useful to us. We can change the gallons to cups by multiplying each gallon amount by \dfrac{16 \text{ cups}}{1 \text{ gallon}}.

Next, let's review Reyita's suggested function modeling the context. After reviewing the accuracy of the table of values with the given context, we can see the structure of the equation is given in the formf \left(x \right) = \dfrac{\text{total cups of salt}}{\text{total gallons of water}}= \dfrac{2x + 22.5}{5x+200}where x represents the time in minutes. This equation is accurate with those units, but using units of cups of salt per gallon of water will not help us solve the main part of the problem. We need to make sure the salinity remains between 1 \% and 3\%. In order to determine the percentage of salt in the water, the volume of the salt and the water needs to be measured with the same units.

To change from gallons of water to cups of water, we multiply the gallons by \dfrac{16 \text{ cups}}{1 \text{ gallon}}. The total gallons of water can be expressed by: 16 \left( 5x + 200 \right) = 80x + 3\,200 by the distributive property. Now we can write a function equation that represents the percentage of salt in the water:f \left(x \right) = \dfrac{\text{total cups of salt}}{\text{total cups of water}}= \dfrac{2x + 22.5}{80x+3\,200}

Finally, we'll change the function and edit the graph to use the converted units and add appropriate labels to the graph:

For each model we should also note the domain the model is appropriate for. See part (b) for considerations of the limitations of the models.

When analyzing models, we should always examine the units to make sure they make sense for the context and produce solutions that are in the desired units.

What are the limitations of our model in terms of the context?

When considering the context of the problem, we should look at aspects such as extraneous solutions or non-viable solutions. Contextual factors may also place constraints on the domain and range.

The domain of the function f\left( x \right) = \dfrac{2x + 22.5}{80x+3\,200} should be restricted from any values where 80x+3\,200=0, so when x=-40. Since x is in minutes, it cannot reasonably be negative and the restriction is not relevant to the context.

The capacity of the aquarium is 500 gallons or 8\, 000 cups, so the time at which the aquarium will reach its maximum amount of water will limit the maximum value of the domain. And since time cannot be a negative value, the domain of the function starts at time 0.

8\,000 \text{ cups} - 3\,200 \text{ cups at time zero} = 4\,800 \text{ cups of water} \\ 4\,800 \text{ cups of water} \div 80 \text{ cups per minute} = 60 \text{ minutes}

The domain of the model is \left[0, 60 \right].

The range constraints should come from the domain constraints, so the range would be \left[f \left(0 \right), f \left(60 \right) \right]. We will still examine our model to make sure the salinity remains in the safe range between 1 \% and 3 \%, but it is contextually possible to have salinity rates outside of the range.

Graphing the function as a system of equations, we can visualize the domain and range restrictions in this context by examining points of intersection:\begin{cases} y = \dfrac{2x + 22.5}{80x+3\,200} \\ x=60 \end{cases}

We could also restrict the function to the appropriate domain in Geogebra using the command: g\left(x\right)=\text{If}\left(0 \leq x \leq 60, f\left(x\right)\right) or alternatively {g\left(x\right)=\text{Function(function, start } x \text{-value, end } x \text{-value)}} with the following inputs g\left(x\right)=\text{Function}\left(f\left(x\right), 0, 60\right).

The limitations of our model in terms of the context may bring up other problems that we will need to consider when interpreting results and drawing conclusions.

Determine the conclusions that can be drawn from our model. State assumptions that were made, if any.

We might assume that the mixture is safe for the fish initially, but we should check the model and confirm that it stays within an appropriate range.

We will consider the salinity of the water while it is filling and after the aquarium is full. Then, we can determine if we should make adjustments to the salt or water prior to filling the aquarium.

By looking at the graph modeling the context, we can see that the salinity of the water starts below 1 \% and slowly rises into the safe range for the fish. We can also calculate the salinity of the water using our table of values and add a column:

We can see from the table and graph that the water in the tank will be safe for fish sometime just before ten minutes. So, no fish should be added to the tank before ten minutes.

We can algebraically check the salinity once the tank is full, to make sure it is still within safe bounds. f\left( 60 \right) = \dfrac{2 \left(60 \right) + 22.5}{80 \left( 60 \right) +3\,200} = \dfrac{142.5}{8\,000}= 0.0178 \approx 1.8 \%

Since the salinity of the full tank is within safe range in this case, we do not need to make any recommendations to changing the context or model.

If the salinity had not been in the safe range once the aquarium was full, we could have considered adjusting the rate of pouring salt, creating different function models with different rates, and determining which model keeps the salinity between 1 \% and 3 \% when the tank is full.

Examine the equation, graph and table that model this context. Are they all accurate models of the situation? If so, explain how the model matches the context. If not, identify any problems with the model, and revise the model to address the problems.

The table created by Maelys is used to construct the equation, so we want to make sure the units of the table are aligned and make sense in context. Then, we should follow up with the units of the equation and determine the accuracy of the graph.

Maelys sets up a table of values for distance, rate, and time in order to use the formula \text{Distance}= \text{Rate} \times \text{Time}. We can assume that if she researched the total distance, that the flight will be nonstop between destinations and that the route is the same there and back. The tailwind will add speed to the flight and the headwind will reduce the speed, so the rate column is accurate.

The formula can be rewritten as \text{Time}=\dfrac{\text{Distance}}{\text{Rate}}, so the entries in the final column of the table are written as \dfrac{\text{Distance}}{\text{Rate}}.

Maelys appropriately writes a math sentence with words to show the equation that she is using to model the speed of the plane, so we can add the time in one direction to the time to fly in the opposite direction for a total of 12 hours and 15 minutes, or 12.25 hours.

The units for the equation also make sense, as it is written as \dfrac{\text{miles}}{\text{miles per hour}}+\dfrac{\text{miles}}{\text{miles per hour}} = \text{hours}

Finally, Maelys rewrites the equation as T \left(x \right)= \dfrac{2\,450}{x+32} + \dfrac{2\,450}{x-32} - 12.25 and the graph has appropriate labels on the axes and for its title.

Subtracting the 12.25 creates a function where the x-intercepts of the graph will help us find the speed of the plane for a 12.25-hour round trip.

Determine the average speed of the plane if the total trip time is 12 hours and 15 minutes, and interpret the results.

As we discussed in the previous part, the function model T \left(x \right)= \dfrac{2\,450}{x+32} + \dfrac{2\,450}{x-32} - 12.25 can be used to find the speed of the plane for a 12.25-hour trip.

We must exclude x=-32 and x=32 from the domain. Since x=-32 is negative, it is not relevant to the problem since negative speeds would not make sense. However, since x=32 will lead to an undefined solution, we must exclude it from the solution as it is an asymptote.

Let's find the x-intercepts of the function in Maelys' graph to determine the speed of the plane. We can write the equation as \dfrac{2\,450}{x+32} + \dfrac{2\,450}{x-32} - 12.25 = 0 and solve for x.

We can use the quadratic formula to find the value of x:

-2.5 miles per hour is an extraneous solution because the plane cannot have a negative speed. So, this intercept is non-viable in context. For the plane to make a 12.25-hour round trip, it will fly at an average speed of 402.5 \text{ mi/hr}.

Revise the model to be useful for comparing any trip times within the given range. Describe any limitations or assumptions in the model.

We need to consider potential contextual and mathematical limitations on the domain and range.

In this context, it does not make sense to talk about negative speeds or negative time. We also do not need to consider speeds that will cause the plane to fly more than 12 hours and 15 minutes, which will be the highest end limit of our range.

If we examine the graph between 0 and 32 \text{ mi/hr}, which is an excluded value of our domain due to its asymptote, we can see that the time, T \left( x \right), has negative values, which does not make sense in our context. We need to examine what is happening at lower speeds of the plane.

An internet search shows that a commercial plane does not take off from the ground until it reaches between 160 and 180 \text{ mi/hr}. So, speeds below 160 \text{ mi/hr} should not be considered part of our domain.

We need a model that will look at speeds for trip lengths between 10 hours and 12.25 hours. We could change her function model to a system of equations.\begin{cases} y = \dfrac{2\,450}{x+32} + \dfrac{2\,450}{x-32} \\y=12.25 \\ y=10 \end{cases}

The graphical model that Maelys uses also limits the length of the trip. We could change Maelys' graph and graph the system of equations instead, so that we can determine the speed of the plane given different time estimates, as follows:

By adjusting Maelys' model, this system of equations and graph would model any specified trip times. We already know the slowest speed will be 402.5 \text{ mi/hr} for a 12.25-hour trip. For a 10-hour hour trip, using the graph to find the intersection point, we get a speed of 492.1 \text{ mi/hr}. Therefore, the domain is \left[402.5, 492.1 \right] with a corresponding range of \left[10, 12.25 \right].

This model will give us approximate values to the actual speeds and times, since it doesn't account for take-off and landing times or time on the ground, and assumes a constant distance of 2\,450 miles in each direction, and a constant wind speed in both directions.

This is not a complete solution for the limitations on the model, but examples that can be taken into consideration when attempting to model a real world context with mathematics.