The cost C in dollars of operating a certain...

The cost $C$ in dollars of operating a certain concrete-cutting machine is related to the number of minutes $n$ the machine is run by the function $C(n)=2.2 n^{2}-66 n+655$ For what number of minutes is the cost of running the machine a minimum? What is the minimum cost?

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Key Concepts

Optimization in Quadratic Functions

Optimization in the context of quadratic functions involves finding the minimum or maximum value of the function, often by using the vertex formula or calculus techniques such as setting the derivative to zero. This approach is useful in real-world applications, such as cost minimization or maximizing efficiency, where determining the optimal value is crucial for decision making.

Parabolic Vertex

The vertex of a parabola represented by a quadratic function is the point where the function attains its maximum or minimum value. When the coefficient a is positive, the parabola opens upwards and the vertex represents the minimum point; when a is negative, the vertex is the maximum point. The x-coordinate of the vertex can be found using the formula -b/(2a), and substituting this back into the function gives the corresponding extremum value.

Quadratic Functions

A quadratic function is a second-degree polynomial function that can be expressed in the form ax² + bx + c, where a, b, and c are constants and a is non-zero. These functions are characterized by their parabolic graphs, which can either open upward or downward depending on the sign of the coefficient a. Quadratic functions are commonly used in various optimization problems, where one is required to find maximum or minimum values.

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