1. A curve is given by the following parametric equations. x = 20 cost, y = 10 sint. The parametric equations are used to represent the location of a car going around the racetrack. a) What is the cartesian equation that represents the race track the car is traveling on? b) What parametric equations would we use to make the car go 3 times faster on the same track? c) What parametric equations would we use to make the car go half as fast on the same track? d) What parametric equations and restrictions on t would we use to make the car go clockwise (reverse direction) and only half-way around on an interval of [0, 2?]? e) Convert the cartesian equation you found in part “a” into a polar equation? Plug it into Desmos to check your work. You must solve for “r”, so “r = ?”

Question
1. A curve is given by the following parametric equations. x = 20 cost, y = 10 sint. The parametric equations are used to represent the location of a car going around the racetrack. a) What is the cartesian equation that represents the race track the car is traveling on? b) What parametric equations would we use to make the car go 3 times faster on the same track? c) What parametric equations would we use to make the car go half as fast on the same track? d) What parametric equations and restrictions on t would we use to make the car go clockwise (reverse direction) and only half-way around on an interval of [0, 2?]? e) Convert the cartesian equation you found in part “a” into a polar equation? Plug it into Desmos to check your work. You must solve for “r”, so “r = ?”

Answers (1)

2021-03-06
Step 1 Answer for sub question a: Given x = 20 cos t, y = 10 sin t To find the cartesian equation that represents the race track the car is going on solve for t using the equation of y. \(y = 10 sin t\)
\(\frac{y}{10} sin t\)
\(t = sin^{−1} (\frac{y}{10})\) Further substitute the value of t in the equation of x. \(x = 20 cos (sin^{−1} \frac{y}{10})\) Thus we have found the cartesian equation that represents the race track the car is going on. Step 2 Answer for sub question b: The parametric equations we would use to make the car go three times as faster on the same track is \(x(t)=20 cost(3t), y(t)=10 sin(3t).\) Step 3 Answer for sub question c: The parametric equations we would use to make the car go half as fast on the same track is \(x = 20 cos (\frac{t}{2}), y = 10 sin (\frac{t}{2})\)
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Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
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1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?
2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
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6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
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