Part I Scenario 1 - Market Analysis The manager of Shooters, a small billiards hall, has been recording various data over the past year in an effort to improve his business. There is not much competition in the local area, so he has had some flexibility with the prices that he charges. His objective over the past year was to determine if there was a connection between the price he charges. the amount of customers that visit the hall each week, and his weekly profit. The data below represents the managers record over the past year. For each week during the year, he recorded the price it would cost to play pool for 1 hour and the average number of hourly patrons during that week. Use this data to determine the weekly demand d as an exponential function of the hourly price p. Part II Week Price, p People, d Week Price, p People, d Week Price, p People, d 1 $ 10.75 47 19 $ 7.75 84 37 $ 6.00 122 2 $ 9.00 78 20 $ 6.75 108 38 $ 7.50 84 3 $ 9.00 67 21 $ 10.75 53 39 $ 9.75 57 4 $ 10.25 63 22 $ 10.50 49 40 $ 7.00 98 5 $ 8.50 86 23 $ 5.50 128 41 $ 9.25 67 6 $ 9.75 65 24 $ 8.25 76 42 $ 7.50 87 7 $ 8.00 81 25 $ 6.50 115 43 $ 9.50 72 8 $ 6.25 118 26 $ 8.50 68 44 $ 6.25 100 9 $ 6.00 123 27 $ 5.00 128 45 $ 10.50 67 10 $ 10.75 61 28 $ 5.75 115 46 $ 9.50 57 11 $ 7.25 100 29 $ 8.75 82 47 $ 8.00 78 12 $ 10.50 56 30 $ 5.50 125 48 $ 9.75 73 13 $ 6.25 102 31 $ 8.75 65 49 $ 7.75 92 14 $ 9.50 61 32 $ 8.50 85 50 $ 8.50 85 15 $ 10.00 63 33 $ 10.00 69 51 $ 6.75 104 16 $ 6.00 105 34 $ 6.75 96 52 $ 5.75 122 17 $ 10.25 57 35 $ 10.00 58 18 $8.50 78 36 $7.25 97 From to day-to-day operations, the manager has determined that he would like to predict the number of customers according to the relationships =20p 0.65, where p is the hourly supply price. What price and quantity would result in Shooters reaching a market equilibrium? Part III Part IV In addition to finding the equilibrium, the manager would like to determine the weekly cost, revenue and profit. Use the above information to determine the revenue R as a function of the hourly price p. Find the derivative of the revenue function and use it to determine the hourly price he should charge in order to maximize revenue. He has estimated that his weekly expenses are about $250 plus about $1.25 per customer. Determine the weekly cost C as a function of the number of patrons q. Combine this with the demand equation to write the weekly cost C as a function of the hourly price p, then determine the weekly profit as a function of p. Find any break-even points and interpret the significance of each. Part V Find the derivative of the profit function and use it to find the hourly price that will achieve the maximum weekly profit. How many customers would you expect to come to Shooters at that price