Question 1189868: 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 has been modelled by the function N(g)=10+0.2g.
a) Determine a function for f(g)=P(g)N(g).
b) Determine a function for f(g)=C(g)/N(g). How likely are you to receive a free baseball cap if you attend game 5?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **a) Function for f(g) = P(g)N(g)**
To find the function f(g), we simply multiply the functions P(g) and N(g):
f(g) = P(g) * N(g)
f(g) = (25 - 0.1g) * (10 + 0.2g)
Now, we expand the expression:
f(g) = 25 * 10 + 25 * 0.2g - 0.1g * 10 - 0.1g * 0.2g
f(g) = 250 + 5g - g - 0.02g²
f(g) = -0.02g² + 4g + 250
Therefore, the function for f(g) is **f(g) = -0.02g² + 4g + 250**. This function represents the total revenue from ticket sales in hundreds of dollars.
**b) Function for f(g) = C(g)/N(g) and Likelihood of Receiving a Free Cap**
To find the function f(g), we divide the function C(g) by the function N(g):
f(g) = C(g) / N(g)
f(g) = (2 - 0.04g) / (10 + 0.2g)
This function represents the ratio of free caps given away to the number of fans in attendance. To find the likelihood of receiving a free cap at game 5, we substitute g = 5 into the function:
f(5) = (2 - 0.04 * 5) / (10 + 0.2 * 5)
f(5) = (2 - 0.2) / (10 + 1)
f(5) = 1.8 / 11
f(5) ≈ 0.1636
To express this as a percentage, we multiply by 100:
0.1636 * 100 ≈ 16.36%
Therefore, the likelihood of receiving a free baseball cap if you attend game 5 is approximately **16.36%**.
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