Question

The maximum value of $z$ in the following equation $z=6xy+y^2$, where $3x+4y\le 100$ and $4x+3y\le 75$ for $x\ge 0$ and $y\ge 0$ is _________

Answer 1

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

In the question, a function $z=6xy+y^2$ is given, and the constraints $3x+4y\le 100$, $4x+3y\le 75$, $x\ge 0$, 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 $(0,25),(0,0)$ and $\left(\frac{75}{4},0\right)$.

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

Since, the critical points are $(0,25),(0,0)$ and $\left(\frac{75}{4},0\right)$.

Evaluate $z$ for $(0,25)$ as follows:

$$\begin{gathered} z=6\left(0\right)\left(25\right)+{\left(25\right)}^2 \\ \Rightarrow z=625 \end{gathered}$$

So, the value of $z$ for $(0,25)$ is $625$.

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

$$\begin{gathered} z=6\left(0\right)\left(0\right)+{\left(0\right)}^2 \\ \Rightarrow z=0 \end{gathered}$$

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

Similarly, Evaluate $z$ for $\left(\frac{75}{4},0\right)$ as follows:

$$\begin{gathered} z=6\left(\frac{75}{4}\right)\left(0\right)+{\left(0\right)}^2 \\ \Rightarrow z=\frac{225}{2} \end{gathered}$$

So, the value of $z$ for $\left(\frac{75}{4},0\right)$ is $\frac{225}{2}$.

Therefore, the maximum value of the given function is $\frac{225}{2}$.

Hence, the answer is $625$.