Final Answer:
The company will maximize their revenue by setting the camera price at $61.50. At this price, the maximum revenue will be $37,987.50.
Step-by-step explanation:
Revenue function: R(p) = -5p^2 + 1230p. This is a quadratic function with a negative leading coefficient, indicating a downward facing parabola.
Maximum revenue: To find the price that maximizes revenue, we need to find the vertex of the parabola. The vertex occurs at the price where the derivative R'(p) = 0.
Differentiate: R'(p) = -10p + 1230.
Set derivative to zero and solve for p: -10p + 1230 = 0 --> p = 123.
Check if it's a maximum: Since the leading coefficient is negative, the derivative being zero corresponds to a maximum.
Maximum revenue: Substitute p = 123 back into the original equation: R(123) = -5(123)^2 + 1230(123) = $37,987.50.
Therefore, the company should set the camera price at $61.50 to achieve their maximum revenue of $37,987.50.