To solve the linear programming (LP) problem presented in the image, follow these steps:

Objective:

Minimize $c = 2x + 4y$

Subject to the constraints:

1. $0.8x + 0.8y \geq 8$
2. $5x + 10y \geq 70$
3. $x \geq 0$, $y \geq 0$

Steps:

1. Rewrite the constraints for simplicity:

   • Constraint 1: $x + y \geq 10$ (dividing by 0.8).
   • Constraint 2: $x + 2y \geq 14$ (dividing by 5).

2. Identify the feasible region:

   • The constraints define a region in the first quadrant ($x, y \geq 0$).
   • The inequalities are transformed into lines:
     • Line 1: $x + y = 10$
     • Line 2: $x + 2y = 14$
   • The intersection points of these lines and the axes will help define the feasible region.

3. Find intersection points:

   • Intersection of $x + y = 10$ and $x + 2y = 14$: Solve the system of equations: $x + y = 10 \quad \text{(1)}$ $x + 2y = 14 \quad \text{(2)}$ Subtract equation (1) from equation (2): $y = 4$ Substitute $y = 4$ into $x + y = 10$: $x = 6$ So, the intersection is $(6, 4)$.

   • Intersection of $x + y = 10$ with $y = 0$ (x-axis): Set $y = 0$: $x = 10$ So, the point is $(10, 0)$.

   • Intersection of $x + 2y = 14$ with $y = 0$: Set $y = 0$: $x = 14$ So, the point is $(14, 0)$.

4. Evaluate the objective function $c = 2x + 4y$ at each vertex:

   • At $(10, 0)$: $c = 2(10) + 4(0) = 20$
   • At $(6, 4)$: $c = 2(6) + 4(4) = 12 + 16 = 28$
   • At $(14, 0)$: $c = 2(14) + 4(0) = 28$

5. Determine the minimum value:

   • The minimum value of $c$ occurs at $(10, 0)$, where $c = 20$.

Solution:

• Minimum value $c$: $20$
• Point where it occurs: $(x, y) = (10, 0)$
• Feasible region: Bounded and connected by the vertices $(10, 0)$, $(6, 4)$, and $(14, 0)$.

Let me know if you'd like further clarification or details on any step! Here are some related questions:

1. How do you determine if an LP problem is unbounded?
2. What are the implications of a feasible region being empty?
3. How do you graphically solve LP problems with three variables?
4. What are some methods for solving LP problems algebraically?
5. How does the Simplex Method compare to graphical solutions for LP?

Tip: When solving LP problems, always check for redundancy in constraints to simplify computations.