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(3x + y) <= (7x + 2y) <= 9x >= 0 and y >= 0, find maximum for P = 2x + y.

a. p=3
b. p=4
c. p=6
d. p=10

User Longwen Ou
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Final answer:

To maximize P=2x+y under the given inequalities, solve the inequalities to find x=y and then use the condition 2x+y ≤ 9x to determine that x can be as large as 3. Therefore, the maximum value of P is 3, as P=3x when x=y.

Step-by-step explanation:

To find the maximum for P = 2x + y given the inequalities (3x + y) ≤ (7x + 2y) ≤ 9x ≥ 0 and y ≥ 0, we need to solve the system of inequalities step by step.

First, we can simplify the given inequalities:

  • 3x + y ≤ 7x + 2y simplifies to 4x ≥ y (after subtracting 3x and y from both sides).
  • 7x + 2y ≤ 9x simplifies to 2x ≥ 2y or x ≥ y (after dividing by 2).
  • The condition x ≥ 0 and y ≥ 0 are already simplified.

Now, we need to find the maximum value of P = 2x + y given these conditions. Since x ≥ y and 4x ≥ y, the maximum value of 2x + y would be when x is as large as possible but still within the constraints. This occurs when x is at its maximum possible value, which is when x = y. We know x = y must be true for the maximum value of P because if y were less than x, then P would not be at its maximum, since we'd have 4x > y and we could increase y (and thus increase P) without violating any of the constraints.

The maximum value occurs when x = y, meaning we have the system:

  • x = y
  • 2x + y ≤ 9x
  • x ≥ 0
  • y ≥ 0

Plugging x = y into 2x + y, we get 3x. To maximize x, given 3x ≤ 9x, we see that x could be as large as 9x/3, which is 3x = 3(1), since x must be ≥ 0. Hence, the maximum value of P is 3.

User Ajay Krishna Dutta
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