Question

The maximum value of $z=5x+3y$, subject to constraints $3x+5y\le 15$, $5x+2y\le 10$ and $x,y\ge 0$ is

• A. $\frac{235}{19}$
• B. $\frac{325}{19}$
• C. $\frac{523}{19}$
• D. $\frac{532}{19}$

Answer 1

The correct option is A

$\frac{235}{19}$

Explanation for the correct option:

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

In the question, a function $z=5x+3y$ is given, and the constraints $3x+5y\le 15$, $5x+2y\le 10$, 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 $(0,0),(2,0),(0,3)$ and $\left(\frac{20}{19},\frac{45}{19}\right)$.

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

Since, the critical points are $(0,0),(2,0),(0,3)$ and $\left(\frac{20}{19},\frac{45}{19}\right)$.

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

$$\begin{gathered} z=5\left(0\right)+3\left(0\right) \\ \Rightarrow z=0 \end{gathered}$$

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

Similarly, Evaluate $z$ for $(2,0)$ as follows:

$$\begin{gathered} z=5\left(2\right)+3\left(0\right) \\ \Rightarrow z=10 \end{gathered}$$

So, the value of $z$ for $(2,0)$ is $10$.

Similarly, Evaluate $z$ for $(0,3)$ as follows:

$$\begin{gathered} z=5\left(0\right)+3\left(3\right) \\ \Rightarrow z=9 \end{gathered}$$

So, the value of $z$ for $(0,3)$ is $9$.

Similarly, Evaluate $z$ for $\left(\frac{20}{19},\frac{45}{19}\right)$ as follows:

$$\begin{gathered} z=5\left(\frac{20}{19}\right)+3\left(\frac{45}{19}\right) \\ \Rightarrow z=\frac{235}{19} \end{gathered}$$

So, the value of $z$ for $\left(\frac{20}{19},\frac{45}{19}\right)$ is $\frac{235}{19}$.

Therefore, the maximum value of the given function is $\frac{235}{19}$.

Hence, option $A$ is the correct answer.