Question

The maximum value of $z$ is where, $z=4x+2y$ subject to constraints $4x+2y\ge 46$ , $x+3y\le 24$ and $x,y\ge 0$, is

• A. $46$
• B. $96$
• C. $52$
• D. None of these

Answer 1

The correct option is B

$96$

Explanation for the correct option:

Step 1: Find the critical points of the given function.

In the question, a function $z=4x+2y$ is given, and the constraints $4x+2y\ge 46$, $x+3y\le 24$, and $x,y\ge 0$ is also given.

Draw a graph describing the given inequalities as follows:

From the graph, it is clear that the critical points are $(9,5),(24,0)$ and $(11.5,0)$.

Step 2: Find the maximum value of the given function.

Since, the critical points are $(9,5),(24,0)$ and $(11.5,0)$.

Evaluate $z$ for $(9,5)$ as follows:

$$\begin{gathered} z=4\left(9\right)+2\left(5\right) \\ \Rightarrow z=46 \end{gathered}$$

So, the value of $z$ for $(9,5)$ is $46$.

Similarly, Evaluate $z$ for $(24,0)$ as follows:

$$\begin{gathered} z=4\left(24\right)+2\left(0\right) \\ \Rightarrow z=96 \end{gathered}$$

So, the value of $z$ for $(24,0)$ is $96$.

Similarly, Evaluate $z$ for $(11.5,0)$ as follows:

$$\begin{gathered} z=4(11.5)+2\left(0\right) \\ \Rightarrow z=46 \end{gathered}$$

So, the value of $z$ for $(11.5,0)$ is $46$.

Therefore, the maximum value of the given function is $96$.

Hence, option $B$ is the correct answer.