Problem:- 1 another type of regression besides the linear and exponential kinds that were covered in class. A town is holding its annual music festival six months from now. Based on last year's ticket sales and survey results of how many tickets could be sold at different prices, a study group presents the table shown below. Use the cost of a ticket in dollars as your xvariable and the estimated overall profit (in thousands of dollars) the town should make at that ticket price as your yvariable.

TicketCost 25 30 35 40 45 50 55 60 OverallProfit 2 17 28 30 34 26 21 7

Complete the following tasks given the information showing your work and explaining.

1. a scatterplot for this data. Include the scatter plot in your solution.

2. Does it seem that there is a linear or an exponential trend? Calculate the correlation coefficient for the linear and the exponential case to answer this question.

3. If you used Excel the scatterplot, choose polynomial under the trendline options. If you used your graphing calculator then instead of using LinReg(ax + b) or ExpReg, use QuadReg to find an equation of the form y = ax2 + bx + c that describes the relationship between the x and y variables. State the values of a, b, and c and the value of the correlation coefficient using four decimal points for each of these values. Such a relationship between x and y is called andquadratic.

4. Write down a paragraph that will contain your conclusion based on the first few parts of this problem. Your writing should explain whether a linear model, expo- nential model or quadratic model was most appropriate in this case.

5. Use your scatterplot with the plotted curve to estimate the values of the xintercept(s) of the quadratic equation that you obtained in part (c) of this problem. What does/these value(s) represent in this context? Remember that an xintercept

is the point where a graph crosses the xaxis.

6. Estimate the ticket cost that would maximize profit for the town. Use a whole number. What is the estimated maximum profit? Round to the nearest thousand. Use your scatterplot to answer this question.

Problem 2: (40 points) Joe is 30 years old today. He has a steady job where he makes $30, 000 a year with a 4% annual raise. He commits himself to spending only 88% of his income each year and investing the other 12%. He puts his investments into a balanced portfolio that earns an average 9% a year.

2

Answer the following questions showing your work.

1. Find an exponential equation that describes the amount Joe earns x years after age 30. Explain.

2. How much is Joe earning when he is 47 years old?

3. Find the amount of money that Joe is allowing himself to spend when he is 58 years old. Calculate the exact amount.

4. How old is Joe when his expenditures for the year exceed $50, 000? Explain.

5. When saving for retirement a crossover point is the point at which your investments begin to earn more money than the cost of your living expenses. Which of the following statements most accurately describes the meaning of the crossover point?

(a) It's the point where all three graphs cross.

(b) It's the time when you start spending more money than you make.

(c) It's the time when the amount of money you are earning from investments is more than the amount you are earning from working.

(d) It's the time when the pink and blue graphs cross.

(e) It's the time when the money you are earning from your retirement investments

passes the amount of money you are spending.

(f) It's the time when you should quit your job.

(g) It's the time when the total amount you have saved is more than the amount you spend each year.

6)Based on the graphs given above, estimate Joe's age when he reaches his crossover point. Explain.

Problem 3: In 1994, the National Football League introduced a salary cap that limits the amount of money spent on players' salaries. The quadratic model

y = 0.2313x^2 + 2.600x + 35.17

approximates this cap in millions of dollars for the years 1994-2009, where x = 0 represents 1994 , x = 1 represents 1995, and so on (source: www.businessinsider.com). Show your work and explain in answering the following questions:

1. Approximate the NFL salary cap in 2007 to the nearest tenth of a million dollars. 2. According to the model, in what year did the salary cap reach 90 million dollars?

Problem 4: uses the concept that the demand or supply of an item influ- ences the price of the item. More specifically, the table contains price-supply data and price-demand data for corn.

1. Use supply in billions of bushels for x and price per bushel in dollars for y to find the line of best fit for the price-supply data. Draw the scatterplot and the line of best fit through it.

2. Use demand in billions of bushels for x and price per bushel in dollars for y to find the line of best fit for the price-demand data. Draw the scatterplot and the line of best fit through it.

3. The equilibrium point is the point of intersection of these two lines when graphed in the same coordinate system. The equilibrium price is the ycoordinate of the equilibrium point and that price would make supply and demand equal to each other. Set the y expressions of the two lines of best fit that you derived earlier in this problem equal to each other and solve for x. Then plug in to one of the y expressions to find the equilibrium price. down the coordinates of the equilibrium point. Round to two decimal points if needed. Include your work.

Price ($/bu)

2.16 , 2.25 , 2.31 , 2.47 , 2.42 , 2.54

Supply(billion bu)

6.34 , 7.35 , 7.73 , 7.93 ,8.04 ,8.33

Price ($/bu) 2.09

2.11,2.21,2.36,2.35,2.42

Demand (billion bu)

9.99 9.31 8.29 8.02 7.73 7.02