4 5 6 7 8 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. 9 1) 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 and state that function. 10 11 12 13 14 15 16 17 18 140 120 109 Week Price, People Week Price, People Week Price, People 1 $ 6.50 99 19 $ 5.25 119 37 $ 10.50 50 2 $ 5.50 120 20 $ 9.75 72 38 $ 7.00 101 3 $ 6.50 115 21 $ 8.50 86 39 $ 9.50 67 4 $ 6.75 106 22 $ 9.25 69 40 $ 6.75 107 5 $ 9.50 73 23 $ 6.25 116 41 $ 10.75 54 6 $ 10.25 54 24 $ 8.75 70 42 $ 8.25 71 7 $ 7.75 80 25 $ 8.00 79 43 $ 8.75 76 8 $ 8.50 81 26 $ 6.50 101 44 $ 8.50 82 9 $ 5.75 109 27 $ 7.25 91 45 $ 7.75 94 10 $ 9.50 70 28 $ 8.25 81 46 $ 10.00 58 11 $ 10.50 55 29 $ 6.50 102 47 $ 6.50 109 12 $ 8.75 69 30 $ 10.25 60 48 $ 7.75 80 13 $ 10.00 60 31 $ 6.25 116 49 $ 8.25 74 14 $ 5.75 125 32 $ 7.50 85 50 $ 5.00 123 15 $ 7.25 93 33 $ 8.50 69 51 $ 5.00 132 16 $ 9.50 62 34 $ 7.25 96 52 $ 8.75 80 17 $ 9.50 67 35 $ 7.25 92 18 $ 10.25 65 36 $ 8.25 88 19 20 21 22 23 24 25 26 27 Axis Title D 29 30 31 32 5 3 B 11 $ 10.50 12 $ 8.75 13 $ 10.00 14 $ 5.75 15 $ 7.25 16 $ 9.50 17 $ 9.50 18 $ 10.25 D 55 69 60 125 93 62 67 63 E F 29 $ 6.50 30 $ 10.25 31 $ 6.25 32 $ 7.50 33 $ 8.50 34 $ 7.25 35 $ 7.25 36 $ 8.25 G 102 60 116 85 69 96 92 88 H 47 $ 6.50 48 $ 7.75 49 $ 8.25 50 $ 5.00 51 $ 5.00 52 $ 8.75 109 80 74 123 132 80 8 2 2) Based on day-to-day operations, the manager has determined that he would like to predict the number of customers he can serve according to the relationships = 20 p where p is the hourly supply price. What 5 price and quantity would result in Shooters reaching a market equilibrium? 5 7 3) In addition to finding the equilibrium, the manager would like to determine the weekly cost, revenue and B profit. Use the above information to determine the revenue R as a function of the hourly price p and state that 9 function. Find the derivative of the revenue function, state that function, and use it to determine the hourly 0 price he should charge in order to maximize revenue. 1 2 4) He has estimated that his weekly expenses are about $250 plus about $1.25 per customer. Determine the 3 weekly cost cas a function of the number of patrons q. State that function. Combine this with the demand 4 equation to state the weekly cost cas a function of the hourly price p, then determine the weekly profit as a 5 function of p and state that function. Find any break-even points and interpret the significance of each. 6 -7 5) state the derivative of the profit function Pio) 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? -9 50 51 52 53 8 54 4 5 6 7 8 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. 9 1) 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 and state that function. 10 11 12 13 14 15 16 17 18 140 120 109 Week Price, People Week Price, People Week Price, People 1 $ 6.50 99 19 $ 5.25 119 37 $ 10.50 50 2 $ 5.50 120 20 $ 9.75 72 38 $ 7.00 101 3 $ 6.50 115 21 $ 8.50 86 39 $ 9.50 67 4 $ 6.75 106 22 $ 9.25 69 40 $ 6.75 107 5 $ 9.50 73 23 $ 6.25 116 41 $ 10.75 54 6 $ 10.25 54 24 $ 8.75 70 42 $ 8.25 71 7 $ 7.75 80 25 $ 8.00 79 43 $ 8.75 76 8 $ 8.50 81 26 $ 6.50 101 44 $ 8.50 82 9 $ 5.75 109 27 $ 7.25 91 45 $ 7.75 94 10 $ 9.50 70 28 $ 8.25 81 46 $ 10.00 58 11 $ 10.50 55 29 $ 6.50 102 47 $ 6.50 109 12 $ 8.75 69 30 $ 10.25 60 48 $ 7.75 80 13 $ 10.00 60 31 $ 6.25 116 49 $ 8.25 74 14 $ 5.75 125 32 $ 7.50 85 50 $ 5.00 123 15 $ 7.25 93 33 $ 8.50 69 51 $ 5.00 132 16 $ 9.50 62 34 $ 7.25 96 52 $ 8.75 80 17 $ 9.50 67 35 $ 7.25 92 18 $ 10.25 65 36 $ 8.25 88 19 20 21 22 23 24 25 26 27 Axis Title D 29 30 31 32 5 3 B 11 $ 10.50 12 $ 8.75 13 $ 10.00 14 $ 5.75 15 $ 7.25 16 $ 9.50 17 $ 9.50 18 $ 10.25 D 55 69 60 125 93 62 67 63 E F 29 $ 6.50 30 $ 10.25 31 $ 6.25 32 $ 7.50 33 $ 8.50 34 $ 7.25 35 $ 7.25 36 $ 8.25 G 102 60 116 85 69 96 92 88 H 47 $ 6.50 48 $ 7.75 49 $ 8.25 50 $ 5.00 51 $ 5.00 52 $ 8.75 109 80 74 123 132 80 8 2 2) Based on day-to-day operations, the manager has determined that he would like to predict the number of customers he can serve according to the relationships = 20 p where p is the hourly supply price. What 5 price and quantity would result in Shooters reaching a market equilibrium? 5 7 3) In addition to finding the equilibrium, the manager would like to determine the weekly cost, revenue and B profit. Use the above information to determine the revenue R as a function of the hourly price p and state that 9 function. Find the derivative of the revenue function, state that function, and use it to determine the hourly 0 price he should charge in order to maximize revenue. 1 2 4) He has estimated that his weekly expenses are about $250 plus about $1.25 per customer. Determine the 3 weekly cost cas a function of the number of patrons q. State that function. Combine this with the demand 4 equation to state the weekly cost cas a function of the hourly price p, then determine the weekly profit as a 5 function of p and state that function. Find any break-even points and interpret the significance of each. 6 -7 5) state the derivative of the profit function Pio) 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? -9 50 51 52 53 8 54