What is the maximum value of $P=6x+8y$ when the conditions are: $2x+y\le30$; $x+2y\le24$; $x\ge0$, $y\ge0$

Question

Accepted Answer

To determine the maximum value of $ P = 6x + 8y $ under the given constraints, we need to analyze the feasible region defined by the inequalities:

1. $ 2x + y \leq 30 $
2. $ x + 2y \leq 24 $
3. $ x \geq 0 $
4. $ y \geq 0 $

These inequalities form a system of linear constraints that define a polygonal feasible region in the first quadrant of the coordinate plane.

**Step-by-step Explanation:**

1. **Identify the Feasible Region:**
   - The line $ 2x + y = 30 $ intersects the axes at $ (15, 0) $ and $ (0, 30) $.
   - The line $ x + 2y = 24 $ intersects the axes at $ (24, 0) $ and $ (0, 12) $.
   - The feasible region is bounded by these lines and the axes, forming a polygon.

2. **Find the Corner Points:**
   - The intersection of $ 2x + y = 30 $ and $ x + 2y = 24 $ can be found by solving the system:
     $$
     2x + y = 30 \
     x + 2y = 24
     $$
     Solving these equations simultaneously:
     - Multiply the second equation by 2: $ 2x + 4y = 48 $
     - Subtract the first equation from this result: $ 3y = 18 $ $\Rightarrow y = 6$
     - Substitute $ y = 6 $ into $ x + 2y = 24 $: $ x + 12 = 24 $ $\Rightarrow x = 12$
     - Thus, the intersection point is $ (12, 6) $.

- Other corner points are where the lines intersect the axes: $ (15, 0) $, $ (0, 12) $, and the origin $ (0, 0) $.

3. **Evaluate $ P = 6x + 8y $ at Each Corner Point:**
   - At $ (0, 0) $: $ P = 6(0) + 8(0) = 0 $
   - At $ (15, 0) $: $ P = 6(15) + 8(0) = 90 $
   - At $ (0, 12) $: $ P = 6(0) + 8(12) = 96 $
   - At $ (12, 6) $: $ P = 6(12) + 8(6) = 72 + 48 = 120 $

4. **Determine the Maximum Value:**
   - The maximum value of $ P $ is 120, which occurs at the point $ (12, 6) $.

**Conclusion:**

- **Option 2 (120) is correct** because it represents the maximum value of $ P = 6x + 8y $ within the feasible region.
- **Option 1 (60), Option 3 (240), and Option 4 (305) are incorrect** because they do not correspond to the maximum value of $ P $ given the constraints. Specifically, 240 and 305 exceed the possible values calculated from the feasible region, and 60 is less than the maximum value found.

Thus, the correct answer is indeed Option 2, as it accurately reflects the maximum value of the objective function within the constraints provided.