Question

(1 point) 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 sels for p-100-8x dollars per unit and commodity B sells for q = 40-10y dollars per unit. The cost (in dollars) of producing these units is given by the joint-cost function C(x, y)-5xy +5. How much of commodity A and commodity B should be sold in order to maximize profit? Commodity A: Commodity B: units units

Answer 1

Solution:

p = 100-8x;

q = 40-10y

　

P(x,y) = x* (100-8x) + y*(40-10y) - (5xy + 5)

dP/dx = 100 - 16x - 5y =0; or 3.2x+y = 20

dP/dy = 40-20y-5x= 0; or x +4y = 8

Solving this we get

3.2x+y = 20

y = 20-3.2x

Putting in x +4y = 8

x + 4(20-3.2x) = 8

x + 80 - 12.8x = 8

80-8 = 12.8x - x

72 = 11.8x

x = 6.10169

y = 20 - 3.2*6.10169 = 0.47458

　

Hence the optimal profit is for:
Commodity A: 6 units
Commodity B: 0 units