Question

Question is in photo

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,
and
produced at each factory, respectively, and is expressed by the joint cost function:

Answer 1

A) The total cost of producing 150 units at Factory X and 170 units at Factory Y is $486,000. B) The expression for the rate of change of costs with respect to units produced at Factory X is
`\[ (\partial C)/(\partial x) = 14x + 6y \]`. C) The expression for the rate of change of costs with respect to units produced at Factory Y is
`\[ (\partial C)/(\partial y) = 6x + 12y \]`.

A) To find the total cost of production when 150 units are produced at Factory X (x = 150) and 170 units are produced at Factory Y (y = 170), substitute these values into the joint cost function:

`\[ C(x, y) = 7x^2 + 6xy + 6y^2 + 2100 \]\[ C(150, 170) = 7(150)^2 + 6(150)(170) + 6(170)^2 + 2100 \]`

`\[ C(150, 170) = 7(150)^2 + 6(150)(170) + 6(170)^2 + 2100 \]\[ C(150, 170) = 157,500 + 153,000 + 173,400 + 2100 \]\[ C(150, 170) = 486,000`

B) To find the rate of change of costs with respect to units produced at Factory X (dx), take the partial derivative of the cost function with respect to x:

Expression:
`\[ (\partial C)/(\partial x) = 14x + 6y \]`

`\[ (\partial C)/(\partial x) = 14(150) + 6(170) \]\[ (\partial C)/(\partial x) = 2100 + 1020 \]\[ (\partial C)/(\partial x) = 3120 \]`

C) To find the rate of change of costs with respect to units produced at Factory Y (dy), take the partial derivative of the cost function with respect to y:

Expression:
`\[ (\partial C)/(\partial y) = 6x + 12y \]`

`\[ (\partial C)/(\partial y) = 6(150) + 12(170) \]\[ (\partial C)/(\partial y) = 900 + 2040 \]\[ (\partial C)/(\partial y) = 2940 \]`