Question $2$

Given a Cobb$-$Douglas production function $Q=K^{0\mathrm{.}6}{L}^{0\mathrm{.}4}$ and a total cost constraint of $C=wL+rK$ with a budget of $C=1000,$ where $w=10$ and $r=20,$ determine the optimal quantities of capital $(K)$ and labor $(L)$ to maximize the production function. Graph the optimal choice of the firm.

Question $3$

In a production setting with a Leontief production function $Q=$min$(\frac{K}{2},\frac{L}{3})$ and a cost constraint represented by $C=wL+rK$ with a budget of $C=240,$ where $w=8$ and $r=12,$ find the optimal quantities of capital $(K)$ and labor $(L)$ to maximize the production function. Graph the firm's optimal choice.

Question $4$

Using a perfect substitutes production function $Q=2K+2L$ and a total cost constraint given by $C=wL+rK$ with a budget of $C=300,$ where $w=10$ and $r=5,$ determine the optimal quantities of capital $(K)$ and labor $(L)$ to maximize the production function. Graph the firm's optimal choice.