Question

The minimum value of the objective function $Z=2x+10y$ subject to constraints $x\ge 0,y\ge 0,x-y\ge 0,x-5y\le -5$ is

• A. $10$
• B. $15$
• C. $12$
• D. $8$

Answer 1

The correct option is B

$15$

Find the minimum value of the given function

Given : $Z=2x+10y$

Subject to constraints,

$x\ge 0,y\ge 0,x-y\ge 0,x-5y\le -5$

Plot the graph with inequalities,

We can see that the required region is unbounded with vertex at $\left(\frac{5}{4},\frac{5}{4}\right)$.

Therefore the function will have the minimum value at $\left(\frac{5}{4},\frac{5}{4}\right)$.

$\begin{array}{rcl}{Z}_{min}& =& 2\left(\frac{5}{4}\right)+10\left(\frac{5}{4}\right)\\ & =& \frac{10+50}{4}\\ & =& 15\end{array}$

Therefore, the minimum value of the function $Z=2x+10y$ is $15$.

Hence option $\left(B\right)$ is the correct answer.