Step 1 :The problem is asking for the minimum value of a quadratic function. The minimum or maximum of a quadratic function \(ax^2 + bx + c\) occurs at \(x = -b/2a\).
Step 2 :In this case, \(a = 0.4\) and \(b = -280\).
Step 3 :So, we need to calculate \(x = -(-280)/(2*0.4)\) to find the number of machines that minimizes the unit cost.
Step 4 :\(a = 0.4\)
Step 5 :\(b = -280\)
Step 6 :\(x = 350.0\)
Step 7 :Final Answer: The number of machines that must be made to minimize the unit cost is \(\boxed{350}\).