1. **State the problem:** We have a production function $$Q = 12 K^{0.4} L^{0.4}$$ where $K$ is capital and $L$ is labor. The prices per unit are 40 for capital and 5 for labor, with a total budget of 800. We want to find the values of $K$ and $L$ that maximize output $Q$ under the budget constraint. 2. **Write the budget constraint:** $$40K + 5L = 800$$ 3. **Set up the Lagrangian to maximize $Q$ subject to the budget constraint:** $$\mathcal{L} = 12 K^{0.4} L^{0.4} - \lambda (40K + 5L - 800)$$ 4. **Find partial derivatives and set them to zero:** $$\frac{\partial \mathcal{L}}{\partial K} = 12 \times 0.4 K^{-0.6} L^{0.4} - 40 \lambda = 0$$ $$\frac{\partial \mathcal{L}}{\partial L} = 12 \times 0.4 K^{0.4} L^{-0.6} - 5 \lambda = 0$$ $$\frac{\partial \mathcal{L}}{\partial \lambda} = 40K + 5L - 800 = 0$$ 5. **From the first two equations, express $\lambda$ and set equal:** $$\lambda = \frac{12 \times 0.4 K^{-0.6} L^{0.4}}{40} = \frac{12 \times 0.4 K^{0.4} L^{-0.6}}{5}$$ 6. **Simplify and solve for $L$ in terms of $K$:** $$\frac{12 \times 0.4 K^{-0.6} L^{0.4}}{40} = \frac{12 \times 0.4 K^{0.4} L^{-0.6}}{5}$$ Cancel common factors 12 and 0.4: $$\frac{K^{-0.6} L^{0.4}}{40} = \frac{K^{0.4} L^{-0.6}}{5}$$ Multiply both sides by 40 and 5: $$5 K^{-0.6} L^{0.4} = 40 K^{0.4} L^{-0.6}$$ Divide both sides by $K^{-0.6} L^{-0.6}$: $$5 L^{1.0} = 40 K^{1.0}$$ So, $$L = 8 K$$ 7. **Substitute $L = 8K$ into the budget constraint:** $$40K + 5(8K) = 800$$ $$40K + 40K = 800$$ $$80K = 800$$ $$K = \frac{800}{80} = 10$$ 8. **Find $L$:** $$L = 8 \times 10 = 80$$ 9. **Calculate maximum output $Q$:** $$Q = 12 \times 10^{0.4} \times 80^{0.4}$$ Calculate powers: $$10^{0.4} = e^{0.4 \ln 10} \approx e^{0.921} \approx 2.512$$ $$80^{0.4} = e^{0.4 \ln 80} \approx e^{0.4 \times 4.382} = e^{1.753} \approx 5.77$$ So, $$Q \approx 12 \times 2.512 \times 5.77 = 12 \times 14.49 = 173.88$$ **Final answers:** - Capital $K = 10$ - Labor $L = 80$ - Maximum production $Q \approx 174$ Note: The problem's graph suggests $L=120$ for max output, but the math with given prices and budget yields $L=80$ for maximum output under the constraint.