Step 1 :The problem is asking for the price that will maximize the company's revenue. The revenue function is given by \(x \cdot(4080-10 x)\).
Step 2 :To find the maximum of this function, we can take the derivative, set it equal to zero, and solve for \(x\). This will give us the critical points of the function.
Step 3 :We can then test these points to see which one gives the maximum revenue.
Step 4 :The derivative of the revenue function is \(4080 - 20x\).
Step 5 :Setting this equal to zero gives us the critical point \(x = 204\).
Step 6 :Testing this point, we find that the maximum revenue is $416160.
Step 7 :Final Answer: The price \(x\) which will maximize the company's revenue is \(\boxed{204}\).