Question

The maximum value of $z=10x+6y$, subject to constraints $x\ge 0$, $y\ge 0$, $3x+y\le 12$, $2x+5y\le 34$ is

• A. $72$
• B. $80$
• C. $104$
• D. $56$

Answer 1

The correct option is D

$56$

Explanation for the correct option:

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

In the question, a function $z=10x+6y$ is given, and the constraints $x\ge 0$, $y\ge 0$, $3x+y\le 12$, $2x+5y\le 34$ is also given.

Draw a graph describing the given inequalities as follows:

From the graph, it is clear that the critical points are $(0,0),(4,0),(2,6)$ and $(0,6.8)$.

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

Since, the critical points are $(0,0),(4,0),(2,6)$ and $(0,6.8)$.

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

$$\begin{gathered} z=10\left(0\right)+6\left(0\right) \\ \Rightarrow z=0 \end{gathered}$$

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

Similarly, Evaluate $z$ for $(4,0)$ as follows:

$$\begin{gathered} z=10\left(4\right)+6\left(0\right) \\ \Rightarrow z=40 \end{gathered}$$

So, the value of $z$ for $(4,0)$ is $40$.

Similarly, Evaluate $z$ for $(2,6)$ as follows:

$$\begin{gathered} z=10\left(2\right)+6\left(6\right) \\ \Rightarrow z=56 \end{gathered}$$

So, the value of $z$ for $(2,6)$ is $56$.

Similarly, Evaluate $z$ for $(0,6.8)$ as follows:

$$\begin{gathered} z=10\left(0\right)+6(6.8) \\ \Rightarrow z=40.8 \end{gathered}$$

So, the value of $z$ for $(0,6.8)$ is $40.8$.

Therefore, the maximum value of the given function is $56$.

Hence, option $D$ is the correct answer.