Maximum and Minimum Problem

Question

A firm manufactures a commodity at two different factories, Factory X and Factory Y. The total cost (in dollars) of manufacturing depends on the quantities, xx and yy produced at each factory, respectively, and is expressed by the joint cost function:

C(x,y)=4x^2+xy+8y^2+3200

A) If the company's objective is to produce 2,000 units per month while minimizing the total monthly cost of production, how many units should be produced at each factory? (Round your answer to whole units, i.e. no decimal places.)

To minimize costs, the company should produce:

(answer) at Factory X and

(answer) at Factory Y

B) For this combination of units, their minimal costs will be how many dollars. (Do not enter any commas in your answer.)

Answer 1

Hi Mike,

I would start by taking a derivative (d/dx) of C(x,y) as follows

dC / dx = 8 x + y + x y’ + 16 y y’

Then since we only want to manufacture x + y = 2000 units,

y = 2000 - x and y’ = -1

Substituting y and y’ into the expression for dC/dx we get

dC / dx = 8 x + (2000-x) - x - 16 (2000-x) = 22 x - 30,000

To find the minimum of C, we need to set dC/dx = 0, and solve for x

22 x - 30,000 = 0

x = 1364 (x has to be an integer) and y = 2000 - x = 636

I will leave it for you to calculate C (1364, 636). I would recommend to set up an Excel spreadsheet, to calculate the costs for all possible (x, y) pairs, and to draw a graph. Please, reach out, if you need help with that.