Answer to Question #213990 in Calculus for nix

Question

A cobb douglas production function for a new company is given by F(x,y)=50x^3/5y^2/5. Where x represents the units of labor and y represents the units of capital. Suppose the units of labor and the capital cost is $200 and $100 each respectively and the budget constraint is $30,000. Find the maximum production level for this manufacturer.

Answer 1

$$200x+100y=30000=>y=300-2x, x, y\geq 0$$

Then

$$F=F(x)=50x^{3/5}(300-2x)^{2/5}, 0\leq x\leq 150$$

Find the first derivative with respect to $x$

$$F'(x)=(50x^{3/5}(300-2x)^{2/5})'$$

$$=50(\dfrac{3}{5}x^{-2/5}(300-2x)^{2/5}-\dfrac{2}{5}(2)x^{3/5}(300-2x)^{-3/5})$$

$$=10x^{-2/5}(300-2x)^{-3/5}(3(300-2x)-4x)$$

$$=100x^{-2/5}(300-2x)^{-3/5}(90-x)$$

$$F'(x)=0=>100x^{-2/5}(300-2x)^{-3/5}(90-x)=0$$

$$=>x=90$$

Critical numbers: $0, 90, 150$

$F(0)=0$

$F(150)=0$

$$F(90)=50(90)^{3/5}(300-90)^{2/5}=$$

$$=1500(1323)^{1/5}\approx6315.43$$

The maximum production level is $6315.43 and occurs when x=90 units of labor and y=120 units of capital.