Market Analysis The manager of Shooters, a small billiards hall, h...

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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. Please be sure to answer each question in a texbox using complete sentences, with proper mathematical notation and units. k Price, p People, d 1 S 1) The data below represents the managers record over the past year. For each 7.50 81 2 $ 10.00 week during the year, he recorded the price it would cost to play pool for 1 hour 70 and the average number of hourly patrons during that week. Use this data to 3 S 5.50 120 determine the weekly demand d as an exponential function of the hourly price 4 $ 5.75 121 and state that function. 5 $ 8.75 74 6 $ 8.50 70 7 S 7.50 2) Based on day-to-day operations, the manager has determined that he would 83 8 $ 9.75 like to predict the number of customers he can serve according to the 59 relationship s = where p is the hourly supply price. What price and 9 $ 10.75 59 quantity would result in Shooters reaching a market equilibrium? 10 $ 8.75 80 11 S 7.25 86 12 S 9.25 76 2 S 9.25 76 3) In addition to finding the equilibrium, the manager would like to determine 13 $ 10.25 61 the weekly cost, revenue and profit. Use the above information to determine the 14 $ 5.25 134 revenue R as a function of the hourly price p and state that function. Find the 15 $ 8.50 82 derivative of the revenue function, state that function, and use it to determine 16 $ 8.50 80 the hourly price he should charge in order to maximize revenue. 17 $ 7.00 103 18 $ 9.00 77 19 $ 5.50 121 20 $ 6.75 102 4) He has estimated that his weekly expenses are about $250 plus about $1.25 21 $ 6.75 92 per customer. Determine the weekly cost C as a function of the number of 22 $ 9.75 64 patrons q. State that function. Combine this with the demand equation to state 23 S 6.75 99 the weekly cost C as a function of the hourly price p, then determine the weekly 24 $ 8.50 76 profit as a function of p and state that function. Find any break-even points and interpret the significance of each. 25 $ 5.75 123 26 $ 10.00 67 27 $ 10.25 5) State the derivative of the profit function P'(p) and use it to find the hourly 52 price that will achieve the maximum weekly profit. How many customers would 28 $ 7.25 89 you expect to come to Shooters at that price? 29 $ 10.75 61 30 S 5.50 118 31 $ 8.75 83 32 S 7.75 86 33 $ 5.00 127 34 $ 10.25 58 35 $ 8.50 77 36 $ 10.50 49 37 $ 9.75 64 Scenario 3 - Price Analysis is last scenario, the manager of Shooters wants to know how senstive the local market will be if he chan ourly price, i.e., if he changes the price a little will he see a large change in the number of customers or ge at all? Such results can have a large impact on revenue. e the demand equation d(p) from Scenario 1 to determine the elasticity E as a function of price p.

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