Question

The coordinate of the point at which minimum value of $z=7x-8y$ subject to constraints $x+y-20\le 0,y\ge 5,x\ge 0,y\ge 0$ is attained, is:

• A. $(20,0)$
• B. $(15,5)$
• C. $(0,5)$
• D. $(0,20)$

Answer 1

The correct option is D

$(0,20)$

Explanation for the correct option:

Finding minimum value of $Z$

Given objective function :

$z=7x-8y$

Subjected to,

$$\begin{gathered} x+y-20\le 0 \\ y\ge 5 \\ x\ge 0 \\ y\ge 0 \end{gathered}$$

Now consider these constrains like a equation and graphing it simultaneously

Here, the corner points as $\left(0,20\right)$, $\left(0,5\right)$ and $\left(15,5\right)$

Now checking the value of $Z$ at every point

At $\left(0,20\right)$$\Rightarrow Z=7\times 0-8\times 20=-160$

At $\left(0,5\right)\Rightarrow Z=7\times 0-8\times 5=-40$

At $\left(15,5\right)\Rightarrow Z=7\times 15-8\times 5=65$

Thus, $Z$ is minimum at $\left(0,20\right)$

Hence, option (D) is the correct answer.