Final answer:
The maximum possible effectiveness score of the quadratic equation E(x) = 2/3x - 1/90x^2, representing the effectiveness of a television commercial, is calculated using the vertex formula and found to be 9.
Step-by-step explanation:
The question asks for the maximum possible effectiveness score of a television commercial, which is represented by a quadratic equation: E(x) = \(\frac{2}{3}x - \frac{1}{90}x^2\). To find the maximum value of this equation, we need to identify the vertex of the parabola it represents. Since this is a downward-opening parabola (because the coefficient of the x^2 term is negative), the vertex will give the maximum effectiveness score.
To find the vertex, we use the formula for the x-coordinate of the vertex of a parabola, \(x = -\frac{b}{2a}\), where a is the coefficient of the x^2 term and b is the coefficient of the x term. Here, a = -\(\frac{1}{90}\) and b = \(\frac{2}{3}\), so:
\(x = -\frac{\frac{2}{3}}{2 \cdot -\frac{1}{90}} = -\frac{\frac{2}{3}}{-\frac{1}{45}} = 2 \cdot 45 = 90\)
Now we calculate E(90) to find the maximum effectiveness score:
E(90) = \(\frac{2}{3} \cdot 90 - \frac{1}{90} \cdot 90^2\) = 60 - 90 = 9
Therefore, the maximum possible effectiveness score is 9.