Question

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

made. If x engines are made, then the unit cost is given by the function C(x)=1.1x²-704x+122,347. How many engines must be made to minimize the
unit cost?
Do not round your answer.
Please help!!

Answer 1

so the graph of the Cost equatiion C(x) is the graph of a parabola opening upwards, like a bowl, with its minimum value at its vertex, so if we just find its vertex, we'd have the engines and at what unit cost.

`\textit{vertex of a vertical parabola, using coefficients} \\\\ C(x)=\stackrel{\stackrel{\textit{\small a}}{\downarrow }}{1.1}x^2\stackrel{\stackrel{\textit{\small b}}{\downarrow }}{-704}x\stackrel{\stackrel{\textit{\small c}}{\downarrow }}{+122347} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right) \\\\\\ \left(-\cfrac{ -704}{2(1.1)}~~~~ ,~~~~ 122347-\cfrac{ (-704)^2}{4(1.1)}\right) \implies\left( - \cfrac{ -704 }{ 2.2 }~~,~~122347 - \cfrac{ 495616 }{ 4.4 } \right)`

`\left( \cfrac{ 704 }{ 2.2 }~~,~~122347 - \cfrac{ 495616 }{ 4.4 } \right)\implies \left( \cfrac{ 704 }{ 2.2 }~~,~~122347 - 112640 \right) \\\\\\ ~\hfill~ {\Large \begin{array}{llll} (~\stackrel{ engines }{320}~~,~~ \stackrel{ \textit{unit cost} }{9707}~) \end{array}}~\hfill~`