Question

$\text{The minimum value of}\text{3x + 5y}\text{such that:}$

$3x+5y\le 15$

$4x+9y\le 8$

$13x+2y\le 2$

$x\ge 0,y\ge 0$

$\text{is ........................}$

• A. 0

Answer 1

The correct option is A 0


$\text{Z = 3x + 5y}$

$\text{Let us consider}3x+5y\le 15\text{...(i)}$

$\text{If 3x + 5y = 15}$

$\text{x = 0, y = 3}$

$\text{x = 5, y = 0}$

$\text{Let us consider}4x+9y\le 8\text{...(ii)}$

$x=0,y=\frac{8}{9}$

$\text{x = 2, y = 0}$

$\text{Let us consider}13x+2y\le 2\text{...(iii)}$

$\text{If 13x + 2y = 2}$

$\text{x = 0, y = 1}$

$x=\frac{2}{13},y=0$

$\text{Comparing (ii) and (iii)}$

$\text{4x + 9y = 8 ...(ii)}$

$\text{13x + 2y = 2 ... (iii)}$

$\text{From (ii)}x=\frac{8-9y}{4}$

$\text{Putting in (iii)}$

$\frac{13(8-9y)}{4}+2y=2$

$\text{104 - 117y + 8y = 8}$

$\text{109y = 96}$

$\therefore \text{y = 0.88}$

$\text{Hence, x = 0.018}$

$\text{Checking at corner points:}$

$\therefore Z(0,\frac{8}{9})=3\left(0\right)+5\left(\frac{8}{9}\right)=4.44$

$\text{Z(0.018, 0.88) = 3(0.18) + 5(0.88) = 4.454}$

$Z(\frac{2}{13},0)=3\left(\frac{2}{13}\right)+5\left(0\right)=0.46$

$\text{At (0, 0) Z = 0, so minimum value will be 0.}$