Homework SourseA company produces x units of commodity A and y units of com
A company produces x units of commodity A and y units of commodity B each hour. The company can sell all of its units when commodity A sells for p=80-4x dollars per unit and commodity B sells for q=75-10y dollars per unit. The cost (in dollars) of producing these units is given by the joint-cost function C(x,y)=2xy+2. How much of commodity A and commodity B should be sold in order to maximize profit?
Commodity A:_____________ units?
Commodity B:_____________ units?
Commodity A:_____________ units?
Commodity B:_____________ units?
Solution
Profit = Income - Expenses P(x,y) = (95-8x)x + (30-5y)y - (4xy-2) P(x,y) = -8x^2 + 95x -5y^2 + 30y - 4xy - 2 At any extrema on the curve, both partial derivatives of the profit function are equal to zero. dP/dx = -16x + 95 - 4y 16 x = -4y + 95 x = (-4y +95)/16 dP/dy = -10y + 30 - 4x 10y = 30 -4x 10y = 30 -4((-4y +95)/16) 10y = 30 - 0.25(-4y + 95) 10y = 30 + y - 24.25 9y = 5.75 y = 5.75/9 = 0.63 units/hr x = (-4(0.63) + 95)/16 X = 5.778 units/hr
