R(p)=-2.5p^2+600p
To find the answer to the question, we need to understand what the function R(p) represents. In this case, R(p) represents a quadratic function that gives the revenue (R) in terms of the number of products sold (p). The function is given by R(p) = -2.5p^2 + 600p.
To calculate the answer, we would need to know what specific question you have related to this function. For example, if you want to find the maximum revenue or the number of products sold that would maximize revenue, we would need to determine the value of p that corresponds to the maximum point on the graph of the function.
To find this value, we can use calculus or complete the square method. Here, I will provide an explanation of the calculus method:
1. To find the maximum point, we first need to calculate the derivative of R(p) with respect to p. The derivative gives us the rate of change of the function at any given point.
2. Taking the derivative of R(p) = -2.5p^2 + 600p, we get R'(p) = -5p + 600.
3. To find the critical point, where the derivative is equal to zero, we set R'(p) = 0 and solve for p: -5p + 600 = 0. Solving for p gives p = 600 / 5 = 120.
4. Now, we need to check if this point is a maximum. We can do this by taking the second derivative of R(p) and evaluating it at the critical point. If the second derivative is negative, then the critical point is a maximum.
5. Taking the second derivative of R(p), we get R''(p) = -5. Since R''(p) is a constant value (-5) and it is negative, we can conclude that the critical point p = 120 is indeed a maximum.
Therefore, the number of products sold (p) that would maximize revenue is 120.
Please note that this explanation is based on the given function, and the specifics of the question may vary. If you have a different question or require further clarification, please provide more details.