To solve the linear programming (LP) problem, we use the simplex method or graphical method since it's a two-variable problem. Let's break it down step by step:

Problem Setup

Objective function: $\text{Maximize } p = 3x + 2y$

Subject to constraints: $0.4x + 0.2y \leq 2$ $0.07x + 0.14y \leq 0.7$ $3x + 3y \leq 18$ $x \geq 0, \, y \geq 0$

Step 1: Rewrite the inequalities

Convert each inequality to standard form:

1. $0.4x + 0.2y \leq 2 \quad \text{or equivalently } y \leq 10 - 2x$
2. $0.07x + 0.14y \leq 0.7 \quad \text{or equivalently } y \leq 5 - 0.5x$
3. $3x + 3y \leq 18 \quad \text{or equivalently } y \leq 6 - x$

Step 2: Identify Feasible Region

The constraints $x \geq 0$ and $y \geq 0$ ensure we are in the first quadrant. The feasible region is bounded by the intersection of the lines defined by the inequalities above.

Step 3: Find Corner Points

The corner points of the feasible region are obtained by solving the equations of the lines at their intersections.

1. Intersection of $y = 10 - 2x$ and $y = 5 - 0.5x$: Solve: $10 - 2x = 5 - 0.5x$ $5 = 1.5x \implies x = \frac{10}{3}, \, y = 5 - 0.5\left(\frac{10}{3}\right) = \frac{5}{3}$ Point: $\left(\frac{10}{3}, \frac{5}{3}\right)$

2. Intersection of $y = 5 - 0.5x$ and $y = 6 - x$: Solve: $5 - 0.5x = 6 - x$ $-1 = -0.5x \implies x = 2, \, y = 5 - 0.5(2) = 4$ Point: $(2, 4)$

3. Intersection of $y = 6 - x$ and $y = 10 - 2x$: Solve: $6 - x = 10 - 2x$ $-4 = -x \implies x = 4, \, y = 6 - 4 = 2$ Point: $(4, 2)$

Step 4: Evaluate the Objective Function at Each Corner Point

The feasible region is defined by the corner points $\left(\frac{10}{3}, \frac{5}{3}\right)$, $(2, 4)$, and $(4, 2)$. Evaluate $p = 3x + 2y$ at each point:

1. At $\left(\frac{10}{3}, \frac{5}{3}\right)$: $p = 3\left(\frac{10}{3}\right) + 2\left(\frac{5}{3}\right) = 10 + \frac{10}{3} = \frac{40}{3} \approx 13.33$

2. At $(2, 4)$: $p = 3(2) + 2(4) = 6 + 8 = 14$

3. At $(4, 2)$: $p = 3(4) + 2(2) = 12 + 4 = 16$

Step 5: Optimal Solution

The maximum value of $p$ occurs at $(4, 2)$ with $p = 16$.

Final Answer

The optimal solution is: $p = 16 \, \text{at } (x, y) = (4, 2)$

If you have additional questions or need a detailed explanation, let me know!

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1. How is the feasible region determined geometrically?
2. What would change if one constraint were removed?
3. How does the simplex method handle such problems algebraically?
4. What is the significance of bounded versus unbounded solutions in LP?
5. Can duality in LP provide insights into this problem?

Tip: Graphing the constraints on a coordinate plane helps visualize the feasible region and ensure accurate corner point identification.