Question

The area bounded by $\min \left( \left| x \right|,\left| y \right| \right)=2$ and $\max \left( \left| x \right|,\left| y \right| \right)=4$ is
A.8 sq unit
B.16 sq unit
C.24 sq unit
D.32 sq unit

Answer 1

Hint: To answer this question, we first graph the lines for the given question at $x=\pm 2$ and $y=\pm 2.$ This is the inner boundary of the region whose area is to be found. Then we graph the lines for the equations $x=\pm 4$ and $y=\pm 4.$ This forms the outer boundary for the region and we find that the area is small portions which are symmetrical. Therefore, we find the area of one such region and multiply it by the number of such areas.

Complete step by step solution:
We first need to graph the lines for the equations $x=\pm 2$ and $y=\pm 2.$ This is for the first part of the question which specifies $\min \left( \left| x \right|,\left| y \right| \right)=2$ , which means that the area formed by these lines form the minimum or inner boundary for the area. We plot these 4 lines on the coordinate axes and label them as shown in the figure below.

Now, $\min \left( \left| x \right|,\left| y \right| \right)=2$ means that the points of intersection of these lines taken two at a time, form the corners of regions such that the corner is the minimum or least point for the whole region. The region away from these points and lines are considered. Now we need to plot the lines for the equations $x=\pm 4$ and $y=\pm 4.$ We plot them in the same graph and we obtain the figure as shown below.

Now, $\max \left( \left| x \right|,\left| y \right| \right)=4$ means that the points of intersection of these lines taken two at a time, form the corners of regions such that the corner is the maximum or the largest boundary point for the whole region. The region inside these points and lines are considered. Therefore, taking an intersection of these two regions, we get the resultant region as shown by the grey shaded area.
Here, there are 4 similar areas. Hence, we shall calculate just the area for one such square and multiply it by 4 since they are all identical. To tell the area of the square, consider the triangle on the topmost right corner. It has all sides of the same length. Now let us find the length of one such side. Looking at the coordinates, we can tell that the length of 1 side of the square is 2 units.
We know the area of a square is given by ${{a}^{2}},$ where a is the side length.
Therefore, area of one small shaded region is,
$\Rightarrow \text{Area of one small shaded region=}{{\text{2}}^{2}}=4\text{ sq unit}$
Now, we have 4 such squares. Therefore,
$\Rightarrow \text{Area of the entire shaded region=4}\times \text{4}=16\text{ sq unit}$
Hence, the correct option is B. which is 16 sq unit.

Note: Students need to know the graphing techniques well. Care must be taken while noting the min and max functions. Min function means that it forms the minimum line or point for the given region. Similarly, max means that it forms the maximum line or point for the given region. They also need to know how to calculate the area of a regular polygon given the coordinates.