College Cost Project

By: Nishali Naik and Veena Suthendran (4th period Wilt)

Part A: Research and Function Writing

Cost of Tuition from 2000 to 2010 where year 2000 x=0 for Brown University and University of North Texas

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Brown University linear, exponential, and cubic models

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Graph of All Functions for Brown University

Key

Red: Linear

Blue: Exponential

Green: Cubic


x=number of years from 2000

y=tuition ($)


Link to Desmos Graph: https://www.desmos.com/calculator/3pozg3jewk

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University of North Texas (UNT) linear, exponential, and cubic models

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Graph of All Functions for UNT

Key

Red: Linear

Blue: Exponential

Green: Cubic


x=number of years from 2000

y=tuition ($)


Link to Desmos Graph: https://www.desmos.com/calculator/eyk0pgbrhh

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Part B: Mathematical Analysis

Predicted Tuition using Linear Model

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Predicted Tuition using Exponential Model

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Predicted Tuition using Cubic Model

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Actual Tuition for the schools in 1990-1991 and 2014-2015

Percent Error between calculated value and actual tuition cost

Percent Error with Linear Model

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Percent Error with Exponential Model

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Percent Error with Cubic Model

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Function that best models the cost of tuition for each school

For Brown University the function which best models the cost of tuition is the linear model. This is seen in the percent error which was calculated by taking the predicted value and subtracting the exact value then dividing this all by the exact value. The predicted value using the linear model for 1990-1991 was $24,692.41 and the exact value during this time period was $20, 720; the percent error was about 19.172%. Additionally during 2014-2015 the projected tuition using the linear model was $43, 191.83 and the exact value was $62, 694; the percent error was about 31.107% error. On the other hand during the time periods 1990-1991 and 2014-2015, the percent error using the exponential function was approximately 26.936% and 30.148% error respectively. Additionally, during the same time periods using the cubic model the percent error was about 73.351% and 22.941% respectively. The linear model has the lowest percent error overall for Brown University, which indicates that this is the best model for this school. University of North Texas (UNT)’s tuition is best modeled by the Exponential Model. The predicted value using the exponential model for 1990-1991 was $2, 013.545 and the exact tuition during this time period was $3, 326; the percent error was about 39.460%. Additionally, during the time period 2014-2015, the projected tuition using the exponential model was $10, 836. 579; however the exact tuition was $22, 030; there wa sa 50.810% error. The percent error was significantly smaller compared to the linear and cubic models. During 1990-1991 and 2014-2015 using the linear model there was about a 95.511% error and a 57.400% error respectively. Also when using the cubic model during those same time intervals the percent error was about 73.351% error and 22.941% error respectively. Ultimately the linear model better modeled the cost of tuition for Brown University, while the exponential model better modeled the cost of tuition for University of North Texas or UNT.

One Model that was better consistently

We predicted that the exponential mathematical model would best represent the rise in tuition over the years, however our data did not agree. In regards to Brown University, the linear model was better consistently at predicting the tuition costs for a specific year. This is represented in the percent error which was derived by taking the predicted value and subtracting the exact value then dividing this all by the exact value. The linear model predicted that in 1990-1991 the tuition cost would be 24,692.41 while the actually cost was $20,720. The percent change between these two was about 19%. Between the years 2014-2015 the percent change was about 31%. However, when calculating the percent with the other functions such as with the exponential the percent change between the years 1990-1991 and 2014-2015 were 27% and 31% respectively. Additionally, the percent change between the same two time intervals in regards to the cubic function was about 73% and 23%. The data indicates that overall the linear model seems to show the least percent error, thus the linear model was better with Brown University. This is not the case with UNT (University of North Texas) because the percent error indicates that the exponential model showed the least percent error. During the same two time intervals as mentioned previously, the percent error for the exponential model was about 40% and 50% respectively. On the other hand, with the linear models the percent error was about 95% and 57%, while with the cubic function the percent error was about 356% and 66% respectively. There doesn't seem to be a consensus on the overall model which was consistently better when looking at both school, Brown University and the University of North Texas had different models which were better suited to predicting their tuition.

Tuition 50 years ago

UNT and Brown Tuition data with tuition from 1950; year 2000= 0

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Mathematical Models (Linear, Exponential, and Cubic) for Brown University and UNT including tuition 50 years ago

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Graph of all functions for Brown University including tuition in 1950

Key

Red: Linear

Blue: Exponential

Green: Cubic


x=number of years from 2000

y=tuition ($)


Link to Desmos Graph: https://www.desmos.com/calculator/wobq6ufkxx

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Graph of all functions for UNT including tuition in 1950

Key

Red: Linear

Blue: Exponential

Green: Cubic


x=number of years from 2000

y=tuition ($)


Link to Desmos Graph: https://www.desmos.com/calculator/lfe7zjm9qp

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Difference between original functions and functions including data from 1950

There was a significant difference in the graphs and functions acquired from the first time the data was collected starting from 2000-2010 and from the second time the data was collected starting from 1950. The cubic function seemed to have the most change in both the graphs. The cubic function in the UNT mathematical model was flipped around from where it originally in the function without the tuition cost from 1950. Additionally, the exponential function also showed a great deal of change from the first function to the second function.

Reflection

This project furthered my understanding on the real world application of exponential, linear, and cubic functions. I enjoyed this project because it was interesting and was focused on one aspect of pre-cal that we are learning about in class. In the future, the thing that will most likely stay with me about this project is that there are many different ways to represent change over time. Different types of functions such as cubic, exponential, and linear could skew your perception on that particular change. This project was a little hard to get started, just because using the casio was a little confusing when trying to copy and paste all the different equations. Finally, we figured out that by using Desmos we could enlarge the graph and view the changes better. Also putting all the information on something other than a google doc such as a s'more took a little bit of time, but overall it was not very hard. We spent a significant amount of time on this project to make sure all the math and the analysis made sense. However, once I understood how to find the data, it started to become a little easier. In regards to picking a college to go to or my college decision in the future, I will now pay closer attention to the pricing or tuition of that college. Throughout my project I realized that the tuition of a public state college was significantly lower than that of a private out of state college. In 40 weeks I will remember how I applied functions to the real world in this project. In 40 months, I will probably remember the different methods that a rate may be represented. In 40 years, I will remember the significant difference in the price of a college due to weather it is in state or out of state or weather it is public or private.