find the minimum value of\nc = 10x + 26y\nsubject to the following constraints:\n\\begin{cases}\n x + y…

find the minimum value of\nc = 10x + 26y\nsubject to the following constraints:\n\\begin{cases}\n x + y \\geq 6\n\\\\\n x + 5y \\geq 14\n\\\\\n x \\geq 0\n\\\\\n y \\geq 0\n\\end{cases}\nc = ?

find the minimum value of\nc = 10x + 26y\nsubject to the following constraints:\n\\begin{cases}\n x + y \\geq 6\n\\\\\n x + 5y \\geq 14\n\\\\\n x \\geq 0\n\\\\\n y \\geq 0\n\\end{cases}\nc = ?

Answer

Explanation:

Step1: Find intersection of (x + y = 6) and (x + 5y = 14)

Subtract the first equation from the second: ((x + 5y)-(x + y)=14 - 6)
(4y = 8) → (y = 2). Substitute (y = 2) into (x + y = 6): (x=6 - 2 = 4). So one vertex is ((4,2)).

Step2: Find intersection of (x + y = 6) with (x = 0)

Set (x = 0) in (x + y = 6): (y = 6). Vertex: ((0,6)).

Step3: Find intersection of (x + 5y = 14) with (y = 0)

Set (y = 0) in (x + 5y = 14): (x = 14). Vertex: ((14,0)).

Step4: Evaluate (C = 10x + 26y) at each vertex

  • At ((4,2)): (C = 10(4)+26(2)=40 + 52 = 92)
  • At ((0,6)): (C = 10(0)+26(6)=0 + 156 = 156)
  • At ((14,0)): (C = 10(14)+26(0)=140 + 0 = 140)

Answer:

92