91Ó°ÊÓ

In the world of economics and business, the revenue function is crucial for determining how changes in price and demand impact overall earnings. For a basketball team, the revenue function captures the relationship between ticket sales and ticket price. More specifically, it is the product of the number of tickets sold and the price per ticket.

In our exercise, we have a scenario where changes in ticket prices affect attendance. We define the price reduction variable as \( x \), representing the number of $5 decreases. This leads to a new ticket price calculated as \( 75 - 5x \), and a new attendance given by \( 14000 + 800x \). The revenue function then becomes:

• Price per ticket: \( 75 - 5x \)
• Attendance: \( 14000 + 800x \)

Combining these, the revenue function is: \[ R(x) = (75 - 5x)(14000 + 800x) \]
This equation is pivotal for determining the ticket price that will maximize the team's revenue.

Quadratic equations are an essential part of solving problems involving parabolic relationships. A quadratic equation usually takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In our revenue problem, expanding the revenue function yields the quadratic equation:\[ R(x) = -4000x^2 - 10000x + 1050000 \]
At this stage, the equation includes:

• \( a = -4000 \)
• \( b = -10000 \)
• \( c = 1050000 \)

This step is vital as the negative sign of \( a \) signifies a downward-opening parabola, which will help us find the maximum revenue point. Recognizing our equation as a quadratic function opens up the analytical methods to find the vertex, which directly leads us to the maximum possible revenue.

The vertex of a parabola represents either the maximum or minimum point of a quadratic function, depending on whether the parabola opens upwards or downwards. In the context of the revenue function, the vertex gives us the exact point where the revenue is maximized.

To find the vertex of the quadratic equation \( R(x) = -4000x^2 - 10000x + 1050000 \), we use the formula for the x-coordinate of the vertex:\[x = -\frac{b}{2a} \]
Substitute the values for \( a \) and \( b \):

• \( a = -4000 \)
• \( b = -10000 \)

This results in:\[x = -\frac{-10000}{2 \times -4000} = \frac{10000}{8000} = 1.25 \]
Since \( x \) needs to be an integer in practical scenarios, we analyze \( x = 1 \) and \( x = 2 \) to find realistic ticket prices and maximize the revenue. This identifies the ticket price that yields the highest revenue, showcasing the power of leveraging a quadratic equation's vertex.