Activity 4.1c Mathematical Modeling
Activity 4.1c Mathematical ModelingIntroductionIn this activity you will collect and analyze data in order to make predictions based on that data. You will use both manual and computer methods to record, manipulate, and analyze the data in order to determine mathematical relationships between quantities. These mathematical relationships can be represented graphically and by equations, also known as mathematical models. You will then use the mathematical models to make predictions related to the quantities.EquipmentEngineering notebookPencilComputer with Excel/Google SheetsProcedurePart 1. Determine a mathematical model for shoe size as a function of the height in the piece. Then use the mathematical model to make predictions.Record the shoe size and height for each member of your group.HeightShoe SizeGraph your data points on the grid below such that height is represented on the x-axis and shoe size is indicated on the y-axis. Label the axes and indicate units.Consider the data that you have graphed and answer the following.Would you expect that this data is linear? Explain your answer.If you were to sketch a line-of-best fit, what would be a reasonable y-intercept. That is, where would the line-of-best fit cross the vertical axis? Explain your answer.Using your predicted y-intercept, sketch a line-of-best-fit for your data on the grid above.Using your line-of-best-fit, complete the following.Estimate the slope of your line-of-best-fit (include the appropriate units). Explain the interpretation of the slope in words.Write an equation for your line of best fit.Use Excel/Google Sheets to create a scatter plot of your data and find a trend line.Input the data in tabular form. Be sure to include column headings. You do not need to include the piece color column.Create a scatterplot of the data. Format the axes, label the axes, and title the chart as shown below.Add a linear trend line. Set the intercept to the value you suggested in number 3 and format the trend line to forecast backward to zero cubes and forward to 12 cubes. Display the equation of the trend line on the chart.Use the equation of the trend line to answer the following.Rewrite the equation of the trend line using function notation where M(n) represents mass and n represents the number of cubes.What is the domain of the function? That is, what values of n make sense?What is the range of the function? What is the slope of your trend line? Explain the interpretation of the slope in words.How does the slope of your line-of-best-fit compare to the slope of the trend line? Why is there a difference?Is the trend line a good representation of the relationship between height and shoe size? Justify your answer.Part 2. Find a mathematical model to represent the minimum jump height of a BMX bike as a function of the bike mass. Then use the mathematical model to make predictions. An engineer is redesigning a BMX bike. He is interested in how the mass of the bike affects the height that the bike reaches when the rider “gets air” or jumps the bike off of a ramp. He asked an experienced rider to test bikes of various masses and recorded the following minimum jump heights. Bike Mass(lbm)Minimum Jump Height (in.)1983.519.582.02079.220.577.12174.92273.322.571.02368.123.565.82464.2Use this data to complete each of the following.Create a scatterplot and find a trend line for the data using Excel. Print a copy of your worksheet that includes the following:Table of data Scatterplot with properly formatted axes, axes labels and units, and an appropriate chart titleTrend line and its equation displayed on the scatterplotWrite the equation relating Bike Mass to Minimum Jump Height in function notation. Be sure to define your variable.What is the domain of the function? Explain.What is the range of the function?What is the slope of the line (include units). Is the slope positive or negative? Explain the interpretation of the slope in words.If the engineer designed a bike that weighs 18 pounds, predict the minimum jump height. Give your answer in inches (to the nearest hundredth of an inch) and feet and inches (to the nearest inch). Show your work.If the engineer designed a bike that weighs1 pound, predict the minimum jump height. Give your answer in inches to the nearest hundredth of an inch and feet and inches to the nearest inch. Show your work. Does the predicted height for a one pound bike make sense? Is this function a good predictor for minimum jump heights at all bike masses? Explain. If the minimum jump height of 89.7 inches is recorded, predict the estimated mass of the bike. Show your work.Extend Your LearningAssume that you will build your puzzle cube from 2 cm cubes of solid gold and each cube had a mass of 153 g. Address each of the following.Give a mathematical model (in function notation) that would represent the mass of a puzzle piece depending on the number of gold cubes used in the piece. Define your variables.What would be the mass of a puzzle piece that is comprised of four gold cubes?If gold sells for $60/g, what is the four-cube gold puzzle piece worth?How many gold cubes would you expect to be included in a puzzle piece that weighs 1071 g?ConclusionWhat is the advantage of using Excel for data analysis?What precautions should you take to make accurate predictions?What is a function? Explain why the mathematical models that you found in this activity are functions. Are all lines functions? Explain. ................
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