1. **Problem statement:** Optimize the total cost function $$C = 3x^2 + 5xy + 6y^2$$ subject to the constraint $$5x + 7y = 1952$$. 2. **Method:** Use Lagrange multipliers to solve constrained optimization problems. 3. **Set up the Lagrangian:** $$\mathcal{L}(x,y,\lambda) = 3x^2 + 5xy + 6y^2 - \lambda(5x + 7y - 1952)$$ 4. **Find partial derivatives and set to zero:** $$\frac{\partial \mathcal{L}}{\partial x} = 6x + 5y - 5\lambda = 0$$ $$\frac{\partial \mathcal{L}}{\partial y} = 5x + 12y - 7\lambda = 0$$ $$\frac{\partial \mathcal{L}}{\partial \lambda} = -(5x + 7y - 1952) = 0$$ 5. **Rewrite system:** $$6x + 5y = 5\lambda$$ $$5x + 12y = 7\lambda$$ $$5x + 7y = 1952$$ 6. **Solve for $x$ and $y$ in terms of $\lambda$:** From first equation: $$5\lambda = 6x + 5y$$ From second equation: $$7\lambda = 5x + 12y$$ Multiply first by 7 and second by 5: $$35\lambda = 42x + 35y$$ $$35\lambda = 25x + 60y$$ Subtract second from first: $$0 = 17x - 25y \implies 17x = 25y \implies x = \frac{25}{17}y$$ 7. **Substitute $x$ into constraint:** $$5\left(\frac{25}{17}y\right) + 7y = 1952$$ $$\frac{125}{17}y + 7y = 1952$$ $$\frac{125}{17}y + \frac{119}{17}y = 1952$$ $$\frac{244}{17}y = 1952$$ $$y = \frac{1952 \times 17}{244} = 136$$ 8. **Find $x$:** $$x = \frac{25}{17} \times 136 = 200$$ 9. **Evaluate $C$ at $(x,y) = (200,136)$:** $$C = 3(200)^2 + 5(200)(136) + 6(136)^2$$ $$= 3 \times 40000 + 5 \times 27200 + 6 \times 18496$$ $$= 120000 + 136000 + 110976 = 366976$$ **Final answer:** The production volumes that optimize the cost are $$x=200$$ and $$y=136$$. The minimum total cost is $$366976$$.