EXPLANATION:
We are given the unit cost to produce x number of airplanes as follows;
[tex]C(x)=0.5x^2-150x+26777[/tex]
However, to minimize the unit cost, we need to first take the derivative of the cost function and then find its value at zero.
Thuis is shown below;
[tex]C(x)=0.5x^2-150x+26777[/tex][tex]\frac{d}{dx}=2(0.5)x^{2-1}-1(150)x^{1-1}+0[/tex]
Note that for a derivative, the constant term is always equal to zero. We can now simplify what we have above;
[tex]\frac{d}{dx}=1x^1-150[/tex][tex]\frac{d}{dx}=x-150[/tex]
We now set this equal to zero and simplify;
[tex]x-150=0[/tex]
Add 150 to both sides;
[tex]x=150[/tex]
ANSWER:
Therefore, to minimize the unit cost, 150 engines must be made.
