- Word problem involing the maximum or minimum of a quadratic functlon
A medical equipment industry manufactures $X$-ray machines. The unit cost $C$ (the cost in dollars to make each $X$-ray machine) depends on the number of machines made. If $x$ machines are made, then the unit cost is given by the function $C(x)=0.4 x^{2}-280 x+64,573$. How many machines must be made to minimize the unit cost?
Do not round your answer:
Answer
Final Answer: The number of machines that must be made to minimize the unit cost is \(\boxed{350}\).
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 \(\boxed{350}\).
