1. **Problem Statement:** We are given a production function $$Q = e^{L^{0.15} K^{0.2} M^{0.1}}$$ and a cost function $$C = L^2 + K^2 + M^2$$ with constraints: $$5L + 2K + M \leq 890$$ $$L + K + 3M \leq 385$$ $$L + K + M \leq 265$$ We want to analyze or optimize these functions under the given constraints. 2. **Understanding the Problem:** - $Q$ represents output depending on inputs $L$, $K$, and $M$. - $C$ is the cost function based on the squares of $L$, $K$, and $M$. - The inequalities are constraints on the inputs. 3. **Approach:** - Typically, such problems involve maximizing $Q$ or minimizing $C$ subject to constraints. - Since the user did not specify, we can consider maximizing $Q$ subject to the constraints. 4. **Key formulas and rules:** - The production function is exponential with inputs raised to powers. - Constraints are linear inequalities. - Optimization under constraints can be approached using methods like Lagrange multipliers or numerical optimization. 5. **Intermediate steps:** - Because of the complexity, let's analyze the constraints and feasible region. - The constraints limit the values of $L$, $K$, and $M$. 6. **Example evaluation:** - Check if the point $(L,K,M) = (100,100,50)$ satisfies constraints: - $5(100) + 2(100) + 50 = 500 + 200 + 50 = 750 \leq 890$ ✓ - $100 + 100 + 3(50) = 100 + 100 + 150 = 350 \leq 385$ ✓ - $100 + 100 + 50 = 250 \leq 265$ ✓ - Calculate $Q$ at this point: $$Q = e^{100^{0.15} \times 100^{0.2} \times 50^{0.1}}$$ Calculate each term: $$100^{0.15} = e^{0.15 \ln 100} = e^{0.15 \times 4.605} \approx e^{0.6907} \approx 1.995$$ $$100^{0.2} = e^{0.2 \ln 100} = e^{0.2 \times 4.605} \approx e^{0.921} \approx 2.512$$ $$50^{0.1} = e^{0.1 \ln 50} = e^{0.1 \times 3.912} \approx e^{0.3912} \approx 1.479$$ Multiply: $$1.995 \times 2.512 \times 1.479 \approx 7.41$$ So, $$Q = e^{7.41} \approx 1647.5$$ - Calculate cost $C$: $$C = 100^2 + 100^2 + 50^2 = 10000 + 10000 + 2500 = 22500$$ 7. **Summary:** - The point $(100,100,50)$ satisfies constraints and yields $Q \approx 1647.5$ and cost $C=22500$. - To find optimal values, one would use optimization techniques considering constraints. **Final answer:** The production function and cost function are given with constraints. Evaluating at a feasible point $(L,K,M)=(100,100,50)$ yields output $Q \approx 1647.5$ and cost $C=22500$. Further optimization requires advanced methods.