A soccer stadium has a seating capacity of 15,000 spectators . With a ticket price of $14, the average attendance is 9,500. A survey suggests that for each dollar the ticket price is lowered , the average attendance will increase by 1,000. A function that models revenue in terms of ticket price is :

Options

A. R(x)=x[15,000+1,000(14-x)]
B. R(x)=x[1,000+9,500(14-x)]
C. R(x)=x[9,500+15,000 (14-x)]
D. R(x)=x[9,500 + 1,000 (14-x)]

Answer

To model the revenue in terms of the ticket price, we can follow these steps:

### Step 1: Define the Variables
Let \( X \) be the ticket price in dollars. The initial ticket price is $14.

### Step 2: Determine the Change in Attendance
According to the problem, for every dollar decrease in the ticket price, the attendance increases by 1,000 spectators. If the ticket price is lowered by \( (14 - X) \) dollars, the increase in attendance can be expressed as:
\[ \text{Increase in attendance} = 1000 \times (14 - X) \]

### Step 3: Calculate Total Attendance
The initial average attendance is 9,500. Therefore, the total attendance \( A \) when the ticket price is \( X \) can be calculated as:
\[ A = 9500 + 1000 \times (14 - X) \]
This simplifies to:
\[ A = 9500 + 14000 - 1000X = 23500 - 1000X \]

### Step 4: Define the Revenue Function
Revenue \( R \) is calculated as the product of the ticket price and the number of tickets sold (which is the total attendance):
\[ R = X \times A \]
Substituting the expression for \( A \):
\[ R = X \times (23500 - 1000X) \]

### Step 5: Expand the Revenue Function
Now, we will distribute \( X \):
\[ R = 23500X - 1000X^2 \]

### Step 6: Final Revenue Function
Thus, the revenue function in terms of the ticket price \( X \) is:
\[ R(X) = -1000X^2 + 23500X \]