Solve the Following Model Graphically

Question

See Image for Graph: https://imgur.com/a/AUaQqdf

Maximize Z = 4x₁+ 5x₂

subject to: x1 + 2x₂ ≤ 10

6x₁ + 6x₂ ≤ 36

x₁ ≤ 4

x₁ , x₂ ≥ 0

Answer 1

x1+2x2 =10

6x1+6x2=36

x1 +x2 =6

subtract 3rd equation from 1st to get

x2 =4

x1 = 2

the above 2 boundary lines intersect at (2,4)

check that point as well as (0,5) and (4,0)

z = 4(2) + 5(4) = 8+20= 28

z = 0 + 5(5) = 25

z = 4(4) + 0 = 25

max z = 28 when x1=2 and x2 =4

graph the boundary lines of the inequalities, 2 downward sloping lines with x1 on the horizontal axis and x2 on the vertical axis. maximum point on the vertical axis is (0,5), maximum x1 coordinate on the horizontal axis is (4,0)

they intersect at (2,4) also graph 4x1+5x2 = 28 which intersects (2,4), (0,28/5) and (7,0)

solution has to be in quadrant I given the x1,x2>0 restriction

this problem resembles a comparable economics problme, the production possiblities frontier, although it uses a curve in place of the inequalities rather than two straight lines.