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 enginesmade. If x engines are made, then the unit cost is given by the function () 0.4x2-88x + 15.636. How many engines must be made to minimize the uncostDo not round your answerNumber of airplane engines:

Answer 1

we know that

The function cost is equal to

$C(x)=0.4x^2-88x+15,636$

This is a vertical parabola open upward

The vertex is a minimum

The x-coordinate of the vertex represents the Number of airplane engines, that minimize the unit cost

The y-coordinate of the vertex is the minimum cost

Convert the given equation into vertex form

step 1

Factor 0.4

$C(x)=0.4(x^2-220x)+15,636$

Complete the square

$C(x)=0.4(x^2-220x+110^2-110^2)+15,636$$\begin{gathered} C(x)=0.4(x^2-220x+110^2)+15,636-4,840 \\ C(x)=0.4(x^2-220x+110^2)+10,796 \end{gathered}$

Rewrite as a perfect square

$C(x)=0.4(x-110)^2+10,796$

The vertex is the point (110,10,796)

therefore

The answer is 110 airplane engines