Question

The maximum value of $z=9x+13y$, subject to constraints $2x+3y\le 18$, $2x+y\le 10$, $x\ge 0$ and $y\ge 0$ is

• A. $130$
• B. $81$
• C. $79$
• D. $99$

Answer 1

The correct option is C

$79$

Explanation for the correct option:

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

In the question, a function $z=9x+13y$ is given, and the constraints $2x+3y\le 18$, $2x+y\le 10$, $x\ge 0$, and $y\ge 0$ is also given.

Draw a graph describing the given inequalities as follows:

From the graph, it is clear that the critical points are $(0,6),(0,0),(5,0)$ and $(3,4)$.

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

Since, the critical points are $(0,6),(0,0),(5,0)$ and $(3,4)$.

Evaluate $z$ for $(0,6)$ as follows:

$$\begin{gathered} z=9\left(0\right)+13\left(6\right) \\ \Rightarrow z=78 \end{gathered}$$

So, the value of $z$ for $(0,6)$ is $78$.

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

$$\begin{gathered} z=9\left(0\right)+13\left(0\right) \\ \Rightarrow z=0 \end{gathered}$$

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

Similarly, Evaluate $z$ for $(5,0)$ as follows:

$$\begin{gathered} z=9\left(5\right)+13\left(0\right) \\ \Rightarrow z=45 \end{gathered}$$

So, the value of $z$ for $(5,0)$ is $45$.

Similarly, Evaluate $z$ for $(3,4)$ as follows:

$$\begin{gathered} z=9\left(3\right)+13\left(4\right) \\ \Rightarrow z=79 \end{gathered}$$

So, the value of $z$ for $(3,4)$ is $79$.

Therefore, the maximum value of the given function is $79$.

Hence, option $C$ is the correct answer.