Solve each problem. Maximum Revenue The revenue of a charter...

Solve each problem. Maximum Revenue The revenue of a charter bus company depends on the number of unsold seats. If the revenue in dollars, $R(x)$, is given by $$ R(x)=-x^{2}+50 x+5000 $$ where $x$ is the number of unsold seats, find the number of unsold seats that produce maximum revenue. What is the maximum revenue?

Solve each problem.
Maximum Revenue The revenue of a charter bus company depends on the number of unsold seats. If the revenue in dollars, $R(x)$, is given by
$$
R(x)=-x^{2}+50 x+5000
$$
where $x$ is the number of unsold seats, find the number of unsold seats that produce maximum revenue. What is the maximum revenue?

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Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written in the form ax² + bx + c. They are widely used to model various phenomena across different fields because they can capture relationships where change accelerates or decelerates. The shape of the graph of a quadratic function is a parabola, and its properties allow us to analyze and solve optimization problems.

Vertex of a Parabola

The vertex of a parabola is the point at which a quadratic function reaches its maximum or minimum value, depending on whether the parabola opens downward or upward. For the standard quadratic form, the x-coordinate of the vertex can be determined by -b/(2a), and the corresponding y-coordinate is found by evaluating the function at that x-value. This concept is essential for pinpointing the optimal point in many mathematical and real-world applications.

Optimization in Economics

Optimization in economics involves determining the best outcome, such as maximum revenue or minimum cost, by analyzing the relationships modeled by mathematical functions. When these functions are quadratic, the vertex represents the optimal solution. This technique is a fundamental tool in economic decision-making, helping businesses to optimize various parameters for better performance.

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