Question 1189868
**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%**.