Maximize and minimize Z = 5x + 10y subject to x + 2y ≤ 120

Question

Maximize and minimize \(Z=5 x+10 y\)

subject to \( x+2 y \leq 120\)

\(x+y \geq 60,\, x-2 y \geq 0,\, x, y \geq 0\)

Answer 1

\(x+2 y=120\)  .....(i)

If x = 0 then y = 60(0, 60)

If y = 0 then y = 120(120, 0)

x + y = 60   .......(ii)

If x = 0 then y = 60(0, 60)

If y = 0 then x = 60(60, 0)

x - 2y = 0

x = -2y  ......(iii)

x = 0, y = 0

For point A from eq. (i) & (iii)

x + 2y = 120

x - 2 y = 0 \(\Rightarrow x=2 y\)

\(2 y+2 y=120 \,\,\,\,y=30\)

From equ. (i)

\(x+2 y =120 \)

\(\Rightarrow x+2 \times 30=120 \)

\(\Rightarrow x=60\)

\(A =(60,30) \)

For point B from equ. (ii) & (iii)

x + y = 60

2y + y = 60

3y = 60  y = 20

B = (40, 20)

x + 20 = 60  x = 40

z = 5x + 10 y

\(A(60,30) \Rightarrow z=5 \times 60+10 \times 30=600 \)

\( B(40,20) \Rightarrow z=5 \times 40+10 \times 20=400 \)

\(C(60,0) \Rightarrow z=5 \times 60+10 \times 0=300\)

\(D(120,0) \Rightarrow z=5 \times 120+10 \times 0=600\)

\(z_{\max }=600 \text { from point } A(60,30) \text { to point } D(120,0)\)

\(z_{\min }=300 \text { when } \quad x=60 \quad y=0\)