How many units factory needs to produce to minimizee the weekly production costs?

Question

I got a question here:

A Factory, that produces flat panels, produces two diferent type of models, A and B as well The weeklyy costs of producing x units of modell A and y units of model B is given by:

C(x,y)=8x^2 + 2xy + 8y^2 + 500

The factory must produce a total of 200 flat panels per week, but decides itself the sharing distribution between models A and B.-xy

The factory must produce total of 200 flat panels per week, but anyway decides itself sharing distribution betweeen our models A and B.-xy

a) So Calculate how many units the factory must produce of model A and model B respectiveely to minimizee the weekly production costs.

I think, that I need to find a minimum of C:

C~(x) = 8x^2 + 2x(200-x) + 8(200-x)^2 + 500

Is it wrong? How could I solve this one? Would appreciate your help. Thanks

Answer 1

Your rewrite of $C$ in terms of $x$ only is correct. Now take the derivative with respect to $x$, set to zero, solve for $x$. If you are not in a calculus course, expand the polynomial and complete the square. You have a parabola and you are seeking the vertex. You should get something that looks like $a(x-b)^2+c$. The minimum comes when $x=b$ and the value is $c$.

Answer 2

C~(x) = 4x^2 + 2x(200-x) + 8(200-x)^2 + 5000

C~(x) = 10x^2 - 2800x + 325000

Minimum will be, when 20x-2800=0

x=140