1. **State the problem:** Minimize the cost function $$C = 15x + 25y$$ subject to the constraints: $$4x + 7y \geq 28$$ $$x \geq 0, \quad y \geq 0$$ 2. **Understand the constraints:** The inequality $$4x + 7y \geq 28$$ means the feasible region is the set of points on or above the line $$4x + 7y = 28$$ in the first quadrant (since $$x,y \geq 0$$). 3. **Find the boundary line:** Rewrite the constraint as: $$7y = 28 - 4x$$ $$y = \frac{28 - 4x}{7}$$ 4. **Find intercepts of the constraint line:** - When $$x=0$$, $$y = \frac{28}{7} = 4$$ - When $$y=0$$, $$x = \frac{28}{4} = 7$$ 5. **Identify corner points of the feasible region:** Since $$x,y \geq 0$$ and $$4x + 7y \geq 28$$, the feasible region is above the line segment joining (0,4) and (7,0). 6. **Evaluate the cost function at corner points:** - At $$x=0, y=4$$: $$C = 15(0) + 25(4) = 100$$ - At $$x=7, y=0$$: $$C = 15(7) + 25(0) = 105$$ 7. **Check if cost decreases along the boundary:** The cost function is linear, so minimum occurs at a vertex or along the boundary. 8. **Check the cost at the intersection with axes and beyond:** Since the feasible region is $$4x + 7y \geq 28$$, points below the line are not feasible. 9. **Conclusion:** The minimum cost is $$100$$ at $$x=0, y=4$$. **Final answer:** $$\boxed{C_{min} = 100 \text{ at } (x,y) = (0,4)}$$