Advertising The effectiveness of a television commercial...

Advertising The effectiveness of a television commercial depends on how many times a viewer watches it. After some experiments an advertising agency found that if the effectiveness $E$ is measured on a scale of 0 to $10,$ then $$E(n)=\frac{2}{3} n-\frac{1}{90} n^{2}$$ where $n$ is the number of times a viewer watches a given commercial. For a commercial to have maximum effectiveness, how many times should a viewer watch it?

Advertising The effectiveness of a television commercial depends on how many times a viewer watches it. After some experiments an advertising agency found that if the effectiveness $E$ is measured on a scale of 0 to $10,$ then
$$E(n)=\frac{2}{3} n-\frac{1}{90} n^{2}$$
where $n$ is the number of times a viewer watches a given commercial. For a commercial to have maximum effectiveness, how many times should a viewer watch it?

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Key Concepts

Quadratic Functions

Quadratic functions are second-degree polynomial functions that are typically represented by the form ax² + bx + c. They model parabolic behavior, meaning their graphs are curves that either open upwards or downwards, which makes them ideal for representing situations where a maximum or minimum value is sought.

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its maximum or minimum value, depending on the direction in which the parabola opens. The x-coordinate of the vertex for a quadratic function can be computed using the formula -b/(2a), which is instrumental in finding the optimal point in many practical applications.

Optimization

Optimization is the process of finding the best solution among a set of feasible choices, usually involving the maximization or minimization of a function. In the context of quadratic functions, optimization is straightforward because the vertex of the parabola directly indicates the point of maximum or minimum effectiveness.

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