A firm manufactures a commodity at two different factories. The total cost of manufacturing depends on the quantities x and y supplied by each factory and is expressed by the joint cost function: C = f(x, y) = 2x 2 + xy + 2y 2 + 500 The company’s objective is to produce 200 units, while minimizing production costs. How many units should be supplied by each factory

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Question:

A firm manufactures a commodity at two different factories. The total cost of manufacturing depends on the quantities x and y supplied by each factory and is expressed by the joint cost function: C = f(x, y) = 2x 2 + xy + 2y 2 + 500 The company’s objective is to produce 200 units, while minimizing production costs. How many units should be supplied by each factory

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Answer:Each factory should supply 100 unitsx = 100 unitsy = 100 unitsStep-by-step explanation:Since the total production must be 200 units, then x + y = 200.The cost function can be rewritten as a function of 'x' as follows:[tex]y=200-x\\C=f(x,y) =2x^2+xy+2y^2+500\\C= f(x) = 2x^2+x(200-x)+2(200-x)^2+500\\C= f(x) = x^2+200x+500+2(x^2-400x+40,000)\\C= f(x) = 3x^2-600x+80,500[/tex]The value of 'x' for which the derivate of the cost function is zero, is the production level that minimizes cost:[tex]C= f(x) = 3x^2-600x+80,500\\C'= f'(x)=0 = 6x-600\\x=100\ units\\y = 200 -x = 100\ units[/tex]In order to minimize production costs, each factory should supply 100 units

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