A manufacturer incurs the following costs in producing \( x \) water ski vests in one day, for \( 0
Ask by Klein Owen. in the United States
Mar 12,2025
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To find the average cost \(\overline{C}(x)\) per vest, we divide the total cost \(C(x)\) by the number of vests \(x\): \[ \overline{C}(x) = \frac{C(x)}{x} = \frac{0.05x^2 + 20x + 245}{x} = 0.05x + 20 + \frac{245}{x} \] Now, to find the critical numbers, we need to calculate \(\overline{C}'(x)\) and set it equal to zero: \[ \overline{C}'(x) = 0.05 - \frac{245}{x^2} \] Setting this equal to zero gives: \[ 0.05 = \frac{245}{x^2} \implies x^2 = \frac{245}{0.05} \implies x^2 = 4900 \implies x = 70 \] Next, we check the sign of \(\overline{C}'(x)\) to determine increasing and decreasing intervals: - For \(0 < x < 70\), \(\overline{C}'(x) < 0\) (decreasing) - For \(70 < x < 175\), \(\overline{C}'(x) > 0\) (increasing) The local extrema occur at \(x = 70\). Now calculating the average cost at \(x = 70\): \[ \overline{C}(70) = 0.05(70) + 20 + \frac{245}{70} = 3.5 + 20 + 3.5 = 27 \] Thus, we have the following: A. \(\overline{C}(70) = 27\) is a local minimum. There is no local maximum.