1.14 Function model construction and application

Identify the problem. State any assumptions.

To identify the problem, we need to clarify what the model intends to measure, predict, and/or solve. Some questions that we need to consider are:

• When did the storm begin and end, and how do those times relate to the times when the data was collected?

• How will "most rain accumulated" be defined?

• Is the amount of rain that accumulated a cause for concern?

Since we cannot research this specific storm, assumptions need to be made about when the storm began and ended and how these times relate to the times when the data collected. We also need to make an assumption about what "most rain accumulated" means.

Let's assume that the data was collected shortly after the storm began, and that the storm ended shortly after two hours. More specifically, we will assume the storm begins at time zero, so there were zero inches of rain when the storm started.

The phrase "the time interval when most rain accumulated" assumes there was a time in the storm when the rain accumulated faster than other times during the storm. As we model and analyze the data, we will need to determine when it started raining harder and when the rain lightened up.

After deciding the specific times when we think it started raining harder and when the rain lessened, we will calculate the specific amount of rain that accumulated during this time interval.

Remember that the assumptions we made above and the ones we will make during the modeling process will affect the final result. Others may make different assumptions that will lead to different results.

Create a new model to represent the data.

We were given data in a table, but this format may not help us determine specific times when the rain accumulated faster or slower. Instead, we will begin by plotting the data on a coordinate plane, then use the graph to help us create an equation model.

Plotting the data points results in the following graph:

Looking at the shape of the data, it appears that a cube root function would be a good model since the rain accumulates slowly, then accumulates quickly, then slows down again. We can transform the parent function y=\sqrt[3]{x} until we find an equation that appears to fit the data well.

The amount of rainfall from this particular storm can be modeled by the equation R=0.15\sqrt[3]{x-65}+0.6 where x is time in minutes and R is total rainfall in inches.

Apply the model to find the time interval when most rain accumulated and how much accumulated during that time. Interpret the results.

At this point, we will need to decide when we think the rain started accumulating more than it did at the beginning of the storm and when it starts to slow down again. It is important that we highlight this assumption when stating the results.

Since cube root functions are symmetric about the inflection point in the center of the graph, we should choose an interval such that the middle value of the interval is the center point of the graph.

According to our model, the center of the data is at x=65. It appears that most of the rain started accumulating about 10 minutes before and 10 minutes after this point, which was between 55 minutes and 75 minutes into the storm.

Substituting these values into the equation model from part (b) gives us:

Next, we will subtract the values to determine how much rain accumulated over that time period: 0.9232-0.2768=0.6464

Therefore, about 0.65 inches of rain accumulated between 55 minutes and 75 minutes into the storm.

This tells us that, although the storm lasted around 2 hours in total, it rained the hardest during a 20 minute period in the middle of the storm.

These results are based on an assumption about when the rain accumulated faster and when it slowed down. Others may have assumed a different time interval, larger or smaller, which would have produced different results.

Verify the model works as intended and improve as needed.

To verify the model, we need to determine whether the model answers the question we need it to answer and whether it answers the question accurately. We will also need to discuss the limitations of the model.

The model was designed to determine the interval when most rain accumulated, and it does satisfy this need. However, the model is limited to the data collected on this particular storm. This model cannot be used to analyze rainfall from any storm in general.

This model is also limited to the data collection process used, and data was collected during 15-minute intervals. This means there is a lack of specific data on how rain accumulated during the most intense part of the storm.

Although the shape of the model may not be completely accurate between 60 and 75 minutes into the storm, the results about when most of the rain accumulated is still fairly accurate.

Create a report about the accumulation of rain during the storm.

In our report, we need to describe how the rain accumulated during the storm, specifically noting the interval during which most rain accumulated and how much rain accumulated during that period of time.

We can also research the levels of rain that lead to flooding and include this information when creating our final report.

According to www.britannica.com, rain is classified in the following ways:

• Light - less than 0.1\text{ in.} per hour
• Moderate - between 0.1\text{ in.} and 0.3\text{ in.} per hour
• Heavy - more than 0.3\text{ in.} per hour

So, for the first 55 minutes of the storm, the residents of Pierson, FL experienced moderate rainfall. During the next 20 minutes of the storm, there was heavy rainfall. This 20-minute interval is when majority of the rain accumulated, amounting to about 0.65 inches. After this, there was moderate rainfall for the remaining 45 minutes of the storm.

According to www.floridadep.gov, large amounts of rain over a short period of time or large amounts of rain over a long period of time can cause flooding. However, just over 1 inch of rain that accumulated over a two-hour time span is not a relatively large amount of rain.

Suppose the storm did not stop after two hours, but continued to rain steadily for another hour. By the end of the third hour, a total of 1.452 inches of rainfall had accumulated.

With this new information, determine whether your model would need to be adjusted and if it does need to be adjusted, describe how.

We will need to determine how much rain would have accumulated according to our current model and if this is consistent with the new data. If it is not, we will need to adjust our model.

Using our current model to determine the amount of rain that would accumulate after 3 hours, we get:

This is not consistent with the new data, so we will need to adjust the model. Since the cube root model accurately represents the first two hours of the storm, we can use a piecewise function model and determine a new equation that models the third hour of the storm.

Let's assume "steadily" refers to a linear increase in the amount of rain accumulated. We need to find the equation of the line that passes through the points \left(120, 1.158\right) and \left(180,1.452\right) representing the amount of rain accumulated between 2 and 3 hours of the storm.

Therefore, the piecewise function that models the amount of rain accumulated over the 3 hour storm is R=\begin{cases}0.15\sqrt[3]{x-65}+0.6, & 0\leq x<120 \\0.0049x+0.57, & x\geq 180\end{cases}

To check how well our piecewise function models the known data, we can graph it with the known data values.

The equations appear to model the data well. As more data is collected, models will often need to be adjusted so they more accurately represent the context.

Create a model to represent the perimeter and area for each garden type. Define your variables and discuss any constraints.

To help us visualize this situation, a diagram would be a good model to start with. The diagram can help us create equation models which will then help us solve the question in part (b).

Because Kala wants to separate the fruit from the vegetables with a row of fencing, we need to include that part of the fence in our calculations. The perimeter will be the fence around the outside and the fence on the inside.

Let x represent the width of the rectangle which is the same length as the fence in the middle. Let y represent the length. Then, the equation for the perimeter is3x+2y=300 The area function is A\left(x,y\right)=xy

If we solve the perimeter equation for y, we get \begin{aligned}3x+2y&=300\\2y&=300-3x\\y&=150-1.5x\end{aligned} If we substitute 150-1.5x for y in the area formula, we can get a better understanding of the function.\begin{aligned}A\left(x\right)&=x\left(150-1.5x\right)\\&=150x-1.5x^2\end{aligned}

Because x represents a length, we know it must be a positive value. We can determine the upper limit of x by finding when either function is greater than zero. \begin{aligned}150-1.5x&>0\\150&>1.5x\\100&>x\end{aligned} This means 0<x<100.

For the circle, the perimeter can also represent the outside of the circle (the circumference) plus the fencing that goes down the middle.

Let r represent the radius of the circle. Then the perimeter equation is 2\pi r+2r=300. Since this is in one variable, we can solve for r.

The area of the circle can be represented by A=\pi r^2. Since we know the radius, we can substitute and solve this to find the size of the circle that is possible with the amount of fencing available.

With 300\text{ ft} of fencing, a circle with an area of 4\,120.964\text{ ft}^2 can be constructed.

Analyze your models to determine which type of garden Kala should make. Include the dimensions of the garden. Give mathematical and contextual arguments to defend your choice.

To answer this question, we will need to make an assumption about what key features Kala and her family might prefer. A rectangular shape is typical and may be easier to create, or may better fit a plot of land. A circle may waste space, or it may be difficult to plant seeds around the edge of the circle. But maybe Kala isn't concerned about the shape. Maybe she simply wants the shape that has the largest area.

In part (a), we found the area of a circular garden with the amount of fencing available is about 4\,121\text{ ft}^2. Graphing the area function of the rectangular garden can help us make decisions about selecting the shape Kala should choose for the garden.

Finding the vertex algebraically or looking at the graph, we find that the maximum area of a rectangular enclosure is 3\,750\text{ ft}^2. So, the area of the rectangular garden will always be less than the area of the circular garden.

Let's assume that Kala prefers gardening in rows. Although the circular garden has more space, the edges of the circle might end up wasting space which would make the area similar to the area of the rectangle since there is only a difference of a few hundred square feet.

With these factors in mind, Kala should build a rectangular garden. Assuming she wants to maximize the area, we can use the x-value of the vertex of the parabola to find the width of the garden. The width of the garden would be 50\text{ ft}, and the length of the garden would be:

Therefore, the dimensions of the rectangular garden would be 50\text{ ft} by 75\text{ ft}.

Our original diagram and equation models represented the situation accurately. But when we analyzed our models to choose a garden shape, we realized that a graph was also helpful to visualize the situation. Sometimes one type of model alone is not sufficient to find a solution. Multiple models were used to make sense of this problem, including diagrams, function equations, and graphs.

Explain which type of regression model you think best fits the data.

Plotting the data values will give insight about which type of function might best model the data. To plot the values, we need to define the variables. Let x represent the number of the month where March is month 1, April is month 2, etc.

This data could be represented by a few different polynomials, so it is important to defend our choice. Note that the curve does not need to pass through every point, but the model would be more accurate if it included as many points as possible.

The data begins increasing, then decreases around x=6. Therefore, a quadratic function appears to be a good function type for this data.

Another student may have argued that a cubic function would fit this best because the point at x=1 appears to be a turning point. Since there is also a turning point around x=6, this must be at least a cubic function.

Someone else may have said a quartic would be best because the data on the right side appears to decrease at a decreasing speed as it would just before a third turning point. Analyzing the data in this way helps us decide which type of model to start with when using technology to find a regression model.

Two previously collected data points were accidentally overlooked when the table was created. The added data points are: \begin{aligned}\text{Jan }1.57\\\text{Feb }1.14\end{aligned} Explain how this affects the type of function you would choose to fit the data.

We had previously defined the independent variable as the number of the month beginning from March. Now, the two months before March have been added, so we need to redefine x. Let x represent the number of the month where January is month 1, February is month 2, March is month 3, etc.

Now, let's re-plot the data values to gain insight on which type of function may be used to model the data.

The added data do affect the type of function we chose in part (a). It is clear that this data now has 2 turning points, so it now appears that a cubic function would fit the data best.

This answer will depend on what type of model was originally chosen in part (a). If a cubic function was already chosen in part (a), then these added data might not affect the decision.

Create a regression model that best fits all the data from the full calendar year. Justify the reasoning for your choice of model.

To create a regression model, we can use a technological tool. To use the tool, we need to enter the given data into a table, then find the polynomial regression of the desired degree.

One technology tool we can use to calculate the regression equation is the statistics calculator in GeoGebra.

1. Enter the x- and y-values in two separate columns, then highlight the data and select Two Variable Regression Analysis:

   

2. Choose Polynomial under the Regression Model drop down menu, then choose 3 under the next drop down menu to find the cubic of best fit:

   

The cubic curve of best fit is okay, but there are several points above and below the curve. We can use Geogebra to try to find a higher degree polynomial that fits the data better. Using the drop down menu on the right, we will choose 4 to examine the quartic curve of best fit.

This curve fits the data much better than the cubic because there are more points on or near the curve.

Therefore, a regression model that can be used to fit the data is f\left(x\right)=0.0027x^4-0.082x^3+0.7866x^2-2.4923x+3.4312.

For this data set, as the degree of the polynomial increases, the curve seems to more accurately model the given data. A 5th degree polynomial would be an even better match.

When finding the best fit for data, statisticians do not just estimate the curve of best fit based on a visual inspection of the graph. Instead, they make other mathematical calculations about their models to quantify how closely the data fits the regression. Statisticians and scientists would also consider features of the both the graph of a function and the context that might lead to choosing a particular model.

The city council wants a report about the rainfall over the course of the year. Explain to the council what information the model created in part (c) can provide. Include any limitations of the model.

The model can be used to fairly accurately evaluate the amount of rainfall that fell in the weeks between data collection points. We could calculate the rainfall at 4.25 months to find out how much rain there was the first week of April, for example.

However, the polynomial function model will not be useful to extend beyond December because a polynomial function will ultimately have end behavior that goes to positive or negative infinity. In other words, the model has domain restrictions from January to December of that particular year. The rainfall for the next year is more likely to repeat a similar pattern to the current year.

Because this data is likely to increase and decrease year after year, a different type of function may be a better fit for modeling multiple years of rainfall. We will learn about other functions that may end up being a better fit for this type of data later in Algebra 2.

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.

Create a function that models Carlee's costs.

We'll need to research the cost of a freeze dryer, candy, packaging, and labels to create a realistic model. Then, we need to define variables and summarize the problem in a way that helps us choose a model.

From some internet searches:

1. A home freeze dryer costs between \$2\,495 and \$5\,090.

2. The cost of candy varies depending on how much is purchased and how it's packaged. For this model, we'll use a 30-pack of individual fruity candy packs that can be purchased for \$25, making each bag about \$0.83.

3. Again, there are a lot of packaging choices out there Carlee can choose from. For this model, we've picked a pink stand-up pouch that's 5.85\text{ in}\times 3.5\text{ in}\times 9.0125\text{ in} and is advertised to hold 6–11 \text{ oz}. Carlee can buy 100 bags for \$42, making the cost of each bag \$0.42.

Carlee only needs to buy the freeze dryer once. Then she will need to spend \$0.83+\$0.42=\$1.25 for each bag of freeze-dried candy she wants to make.

We'll assume that Carlee's costs for candy and packaging won't vary over time which means we can use a linear function to model total costs. Let x represent the number of bags of freeze-dried candy Carlee produces, then C(x)=1.25x+5000 represents her total cost, assuming she buys a more expensive freeze dryer.

Some other assumptions we could consider:

• Carlee's costs are \$1.25 but she should plan \$1.50 to account for any unexpected changes in price: C_1(x)=1.5x+5000

• Carlee purchases a cheaper freeze dryer: C_2(x)=1.25x+3000

• Carlee also needs to pay \$100 for a business license and pays a graphic designer \$150 for a logo: C_3(x)=1.25x+5250

Determine a reasonable selling price and create a function to model Carlee's revenue.

Consider the market price of freeze-dried candy from other sellers, how much Carlee wants to earn from each bag sold (her profit margin), and how long it will take her to pay off the initial costs before she makes a profit.

There are companies online selling freeze-dried candy bags between \$10 and \$12, and when she went to her local farmer's market, Carlee saw someone selling freeze-dried candy for \$8 per bag.

Carlee's revenue function would be R(x)=cx where c is her cost per bag. For now, let's assume a cost of \$10, which is in the middle of what she found. Thus, Carlee's revenue function is R(x)=10x.

Combine your functions from (a) and (b) to create a function that models the business owner's profit.

Profit is the difference between revenue and cost.

Carlee's revenue function is R(x)=10x and her cost function is C(x)=1.25x+5000 so her profit function will be:

P(x)=8.75x-5000

Use your cost, revenue, and profit models to write a business plan for the business owner.

Carlee needs to know when her business will start profiting. We should also present her with a few reasonable options.

We uncovered multiple options for cost and price that impact Carlee's profits, so we'll display a few options for her business plan:

If Carlee purchases the \$5\,000 freeze dryer and sells her candy at \$10 per bag with a \$1.25 cost per bag in materials, she'll need to sell 572 bags to pay off her initial costs and start profiting. If she purchases a cheaper freeze dryer at \$3\,000 and keeps her selling price the same, she can begin profiting after only 343 are bags sold. If Carlee buys the more expensive freeze dryer, increases the cost per bag to \$1.50, and sells the bags at \$12 each, she will need to sell 477 bags to start profiting.

In summary, Carlee should expect to make between 300 and 600 bags of candy before she makes a profit. We recommend that Carlee:

• Purchase the most reliable machine at the cheapest price.

• Take advantage of bulk pricing and plan to buy supplies for making at least 600 bags of candy (to stay out of debt).

• Find the highest selling price point that still gets her the business she needs to succeed.