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=80-5x dollars per unit and commodity B sells for p=75-10y dollars per unit. The cost (in dollars) of producing these units is given by the joint-cost function C(x,y)=4xy+4. 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
sol
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
