Final answer:
To maximize Z=2X₁+3X₂+4X, we solve the given inequalities separately to find the valid ranges for X₁ and X₂. Then, within those ranges, we select the values that will maximize Z.
Step-by-step explanation:
To maximize Z=2X₁+3X₂+4X, subject to X₁+2X₃ ≤ 64 and X₁+2X₂ ≤ 10, we need to find the maximum values for X₁ and X₂ that satisfy the given constraints. Let's solve the two inequalities separately:
- X₁+2X₃ ≤ 64:
Assume X₃ = 0, so X₁ ≤ 64.
Assume X₁ = 0, so 2X₃ ≤ 64,
So the range of X₃ is 0 ≤ X₃ ≤ 32.
Hence, the range of X₁ is 0 ≤ X₁ ≤ 64. - X₁+2X₂ ≤ 10:
Assume X₂ = 0, so X₁ ≤ 10.
Assume X₁ = 0, so 2X₂ ≤ 10.
So the range of X₂ is 0 ≤ X₂ ≤ 5.
Hence, the range of X₁ is 0 ≤ X₁ ≤ 10.
Considering both inequalities, we find that the valid range for X₁ is 0 ≤ X₁ ≤ 10 and the valid range for X₂ is 0 ≤ X₂ ≤ 5. To maximize Z, we need to choose the values of X₁ and X₂ that will give us the highest value for Z within these valid ranges.