Question

If $Z=5x+2y$ subject to the following constraints : $x-2y\le 2,$ $3x+2y\le 12,$ $-3x+2y\le 3,$ $x\ge 0,y\ge 0,$ then the absolute value of the difference of the maximum and minimum value of $Z$ is

[1 mark]

• A. $16$
• B. $19$
• C. $3$
• D. $17$

Answer 1

The correct option is B $19$

The constraints are
$x-2y\le 2\cdots \left(i\right)$
$3x+2y\le 12\cdots \left(ii\right)$
$-3x+2y\le 3\cdots \left(iii\right)$
$x\ge 0,y\ge 0\cdots \left(iv\right)$


It is observed that the feasible region $OBGHJ$ is bounded.
Thus, we use corner point method to determine the maximum and minimum value of $Z,$ where $Z=5x+2y$

$\begin{array}{cc}\text{Corner point}& \text{Corresponding value of}Z\\ B(2,0)& 10\\ G(\frac{7}{2},\frac{3}{4})& 19\\ H(\frac{3}{2},\frac{15}{4})& 15\\ J(0,\frac{3}{2})& 3\\ O(0,0)& 0\end{array}$

Hence, $Z_{max}=19$ at the point $G(\frac{7}{2},\frac{3}{4})$
and $Z_{min}=0$ at the point $O(0,0)$
$\Rightarrow |Z_{max}-Z_{min}|=19-0=19$