Question
A supply company manufactures copy machines. The unit cost \( C \) (the cost in dollars to make each copy machine) depends on the number of machines made. If \( x \) machines are made, then the unit cost is given by the function \( C(x)=1.2 x^{2}-264 x+19,560 \), How many machines must be made to minimize the unit cost? Do not round your answer.
Ask by Floyd Love. in the United States
Feb 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
To minimize the unit cost, the company should manufacture **110 machines**.
Solution
The unit cost function is given by C(x) = 1.2x² - 264x + 19,560. This is a quadratic function where the coefficient of x² is positive, meaning the function has a minimum value.
To find the value of x that minimizes the function, we use the vertex formula for a quadratic function, x = -b/(2a). Here, a = 1.2 and b = -264.
Plugging in the values:
x = -(-264)/(2 × 1.2)
x = 264/2.4
x = 110
Thus, 110 machines must be made to minimize the unit cost.
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Simplify this solution Extra Insights
To minimize the unit cost \( C(x) = 1.2x^{2} - 264x + 19,560 \), you can use the vertex formula of a parabola, which is given by \( x = -\frac{b}{2a} \). Here, \( a = 1.2 \) and \( b = -264 \). Plugging in these values: \[ x = -\frac{-264}{2 \cdot 1.2} = \frac{264}{2.4} = 110 \] Thus, to minimize the unit cost, 110 machines must be made. Also, check the unit cost calculation by evaluating \( C(110) \) for practical insights on production efficiency and cost-effectiveness—a great way to ensure your calculations lead to real-world savings!
