An aircraft factory manufactures airplane engines. The unit cost $C$ (the cost in dollars to make each airplane engine) depends on the number of engines ma $x$ engines are made, then the unit cost is given by the function $C(x)=1.2 x^{2}-216 x+22,120$. How many engines must be made to minimize the unit cost? Do not round your answer.
Number of airplane engines: [1]
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Step 1 ：The problem is asking for the number of engines that must be made to minimize the unit cost. This is a problem of finding the minimum of a function, which can be solved by finding the derivative of the function and setting it equal to zero. The derivative of a function gives the rate of change of the function, and where this rate of change is zero, the function has a local minimum or maximum. In this case, we are looking for a minimum, so we need to find where the derivative of the function is zero.

Step 2 ：The derivative of the function \(C(x)=1.2 x^{2}-216 x+22,120\) is \(C'(x) = 2.4x - 216\). Setting this equal to zero gives the critical point \(x = 90\).

Step 3 ：We need to verify that this is indeed a minimum and not a maximum. We can do this by taking the second derivative of the function and checking its sign at the critical point. If the second derivative is positive, then the function has a minimum at the critical point.

Step 4 ：The second derivative of the function is \(C''(x) = 2.4\). Evaluating this at the critical point \(x = 90\) gives \(C''(90) = 2.4\), which is positive.

Step 5 ：Therefore, the function has a minimum at this point. Therefore, the unit cost is minimized when 90 engines are made.

Step 6 ：The number of engines that must be made to minimize the unit cost is \(\boxed{90}\).