2. Baseball Revenues In an effort to boost fan support, the owners of a baseball team have agreed to gradually reduce ticket prices, P, in dollars, according to the function P(g) = 25 – 0.1g where g is the number of games that have been played so far this season. The owners are also randomly giving away free baseball caps. The number, C, in hundreds of caps given away per game can be modelled by the function C(g) = 2 – 0.04g. Since these marketing initiatives began the number, N, in hundreds, of fans in attendance have been modelled by the function N(g) = 10 + 0.2g
3. The Problem A) Develop an alegbraic and a graphical model for f(g) = P(g)N(g) and explain what it means. Will the owners increase their revenue from ticket sales under their current marketing plan? B) Develop an algebraic and a graphical model for f(g) = C(g) / N(g) and explain what it means. How likely are you to receive a free baseball cap if you attend game 5?
4. Solution A) Multiply P(g) by N(g) to produce the combined function f(g) = P(g)N(g). F(g) = P(g)N(g) = (25 – 0.1g)(10 + 0.2g) = 250 + 5g – g – 0.022 = -0.02g2 + 4g + 250
6. What this all means This combined function is the product of the ticket price and the number of fans attending, in hundreds. Therefore, f(g) = P(g)N(g) represenets the revenue from ticket sales, in hundreds of dollars. Note that the function is quadratic, increasing until about game 100, and then decreasing after that. Therefore, the owners will increase their revenue from ticket sales in the short term under their current marketing strategy, but eventually this strategy will no longer be effective.
7. Cont’d Divide C(g) by N(g) to produce the combined function f(g) = C(g) / N(g) Solution F(g) = C(g) / N(g) = 2-0.04g / 10+0.2g Use the graphing calculator to graph this function. Enter the equations C(g) = 2 – 0.04g and N(g) = 10 + 0.2g first. Then, use them to produce the graph of f(g) = C(g) / N(g). To view only the quotient function, leave it turned on, but turn the other functions off. Apply nnumber sense of systematic trial to set an appropriate viewing window.
8. Problem #2 The combined function f(g) = C(g) / N(g) represents the number of free caps randomly given out divided by the number of fans. Therefore this function represents the probability that a fan will receive a free baseball cap as a function of the game number. To Determine the probability of recieiving a free cap at game 5, evaluate f(g) = C(g) / N(g) for g = 5
9. The Solution Substitute g = 5 into the equation for f(g) = C(g) / N(g) and evaluate. F(g) = 2 – 0.04g / 10 + 0.2g F(5) = 2 – 0.04(5) / 10 + 0.2(5) = 2 – 0.2 / 10 + 1 = 1.8 / 11 = 0.163 Therefore, according to the equation, there is about a 16% chance of receiving a free baseball cap at game 5.
