An Artist Produces Two Items for Sale $R ( x , y ) = 40 x + 7 x y + 80 y ^ { 2 } + 10 y$

Multiple Choice

An artist produces two items for sale. Each unit of item I costs $50 to produce, while each unit of item II costs $200. The revenue function is $R ( x , y ) = 40 x + 7 x y + 80 y ^ { 2 } + 10 y$ , where x is units of item I and y is units of item II. Suppose the artist has only $1000 to spend on production. Which of the following is the Lagrange function the artist should use to determine what combination of production amounts (x, y) will yield maximum profits subject to the constraint that his costs must equal $1000.

A) -10x + 7xy + 80 $y ^ { 2 }$ - 190y + λ(200x + 50y - 1000)
B) 40x + 7xy + 80 $y ^ { 2 }$ + 10y + λ(50x + 200y - 1000)
C) -10x + 7xy + 80 $y ^ { 2 }$ - 190y + λ(50x + 200y - 1000)
D) 40x + 7xy + 80 $y ^ { 2 }$ + 10y + λ(200x + 50y - 1000)
E) none of these