#### Answer

No, the person who is $6$ feet tall and weigh $205$ pounds is not in the healthy weight region.

#### Work Step by Step

It is known that, $1\text{ feet}=\text{12 inches}$
Let us consider height in feet and convert it into inches as given below:
$\begin{align}
& \text{6 feet}=\text{6}\times \text{12 inches} \\
& =\text{72 inches}
\end{align}$
The coordinates which define the height and weight of a person are $\left( 72,205 \right)$. So, in order to check whether this point lies in the healthy weight region or not, substitute the coordinates for x and y variables respectively in both the provided equations as shown below:
Put the values in the first equation as given below:
$\begin{align}
& 5.3x-y\ge 180 \\
& 5.3\left( 72 \right)-205\ge 180 \\
& 176.6\ge 180
\end{align}$
Which is incorrect. Therefore, the inequality does not hold.
Now, put the values in the second equation as given below:
$\begin{align}
& 4.1\left( x \right)-y\le 140 \\
& 4.1\left( 72 \right)-205\le 140 \\
& 90.2\le 140
\end{align}$
Here, the inequality holds.
Therefore, both equations are not satisfied by the coordinates of a person's height and weight, which means that the point does not lie in the healthy weight region.
Hence, the person who is $6$ feet tall weighs $205$ pounds is not in the healthy weight region.