Final answer:
To maximize revenue from selling Awesome Hearing Aids, the revenue function R(x) = -2x² + 115x must be analyzed. The maximum occurs at the vertex of the parabola, which is 29 hearing aids when rounded to the nearest whole number.
Step-by-step explanation:
To find the number of Awesome Hearing Aids that need to be produced and sold to maximize revenue, we need to analyze the revenue function R(x) = -2x² + 115x. Since this is a quadratic function, the maximum revenue occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula -b/(2a), where a and b are coefficients from the quadratic equation ax² + bx + c. In this case, a = -2 and b = 115. Calculating this gives us:
Vertex x-coordinate = -b/(2a) = -115/(2 × -2) = 28.75
Since we need a whole number of hearing aids, we round this to the nearest whole number, resulting in 29 hearing aids. We then substitute this back into the function to find the maximum revenue:
R(29) = -2(29)² + 115(29)
R(29) = -2(841) + 3335
R(29) = -1682 + 3335
R(29) = $1653
However, none of the provided answer options exactly match $1653. Since option c) and d) have the same number of hearing aids but different revenues.