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=45-4x dollars per unit and commodity B sells for 30-8y dollars per unit. The cost (in dollars) of producing these units is given by the joint-cost function C(x,y)=5xy+3. How much of commodity A and commodity B should be sold in order to maximize profit?
Units of Commodity A:
Units of Commodity B:
Units of Commodity A:
Units of Commodity B:
Solution
Profit = Income - Expenses P(x,y) = (45-4x)x + (30-8y)y - (5xy+3) P(x,y) = -4x^2 + 45x -8y^2 + 30y - 5xy - 3 At any extrema on the curve, both partial derivatives of the profit function are equal to zero. dP/dx = -8x + 45 - 5y 8x +5y =45 x= (45-5y)/8 dP/dy = -16y + 30 - 5x 5x +16y =30 5(45-5y)/8 + 16y =30 225 -25y + 128y = 240 103y = 15 y=15/105 = .1456 unit/hr x= (45-5y)/8 x= 5.5910 units/hr so we know that A = x= 5.5910 units/hr and B = y = .1456 unit/hr
