# Math Modeling Page

### Mathematical Modeling at Mass Academy is a highly collaborative course that challenges and improves students' problem-solving skills. Students learn and refine their understanding of new mathematical concepts through daily practice and apply their skills to real-world mathematical modeling problems.

From November 10th to the 15th, all students in MAMS participated in HiMCM, the High School Mathematical Contest in Modeling. HiMCM is an annual competition that gives students the opportunity to work in teams to solve real-life mathematical modeling problems. The HiMCM provides two problems to choose from each year, and my group chose problem B. This problem was about the drought at Lake Mead. We were provided with details about Lake Mead, such as the amount of water sent from Lake Mead to surrounding states annually, the inflow of water to Lake Mead, data on the elevation, volume, and area of Lake Mead, and methods for reusing water. My group chose this problem because of the quantity of data and information we were provided; we were curious about how in-depth we could go with our solution. We had several different questions within the problem: we were asked to analyze inflow, outflow, and evaporation loss of water volume at Lake Mead, define the relationship between the elevation, volume, and area of Lake Mead, develop models to predict water levels at Lake Mead in 2025, 2030, and 2050 based on historical and recent trends, develop a plan to recycle wastewater to prevent further drought, and create a one-page non-technical newsletter to report our findings. We analyzed the relationship between inflow, outflow, and loss by calculating the value of each variable annually. Then, we subtracted inflow and loss from outflow to find the yearly change in water volume at Lake Mead. To find the connection between elevation, volume, and area, we plotted values for each of these variables against each other. After plotting the variables together, we were able to find equations to convert each variable into another. In order to predict water elevation levels at Lake Mead in the future, we created a trendline using historical and recent elevation level data. Using this trendline, we estimated water elevation levels in 2025, 2030, and 2050. While developing a plan to recycle wastewater, we researched the attitudes of each state’s government to preventing water wastage and drought. Additionally, we researched the cost of recycling wastewater to create a reasonable plan to prevent further drought at Lake Mead. Finally, we condensed our takeaways on this issue into a short news article.

For this assignment, we were given a set of data to plot and fit a median-median regression line to. We learned how to create a median-median regression line prior to this project, so this was our chance to implement our knowledge. To start, I created a table and graph of the data. Then, I split the data into three parts, with the first and third parts having an equal amount of values. I then found the median of each of the three parts of the data. These medians are called summary points. I graphed the summary points along with the rest of the data and found the slope of the line that goes through the first and third summary points. Next, I found the y-intercept for this line and the y-intercept for a line of the same slope going through the second summary point. After this, I took the difference between the second y-intercept and the first y-intercept. Next, I determined the median-median regression line, which has the same slope as the previous two lines with a y-intercept that equals the sum of the first y-intercept added to the difference between the two y-intercepts. I graphed my median-median regression line with the rest of the data and determined the distances between the y-coordinate of each data point and the corresponding coordinate on the regression line. These distances are called residuals, and after finding them, I graphed them with the regression line and the data.