Chapter 2.4, Problem 37E

To determine

The value of $\text{r}$ for which the area of the shaded region of the given figure will be greater than or equal to $\text{9(}\pi -2\text{)}$ .

Explanation of Solution

Given information:

The given figure is shown below.

Calculations:

The given figure shows a square is inscribed inside a circle.

The radius of the circle is $\text{r}$ .

Now the area of the circle is $\pi r^2$

Now the length of each side of the square can be found using pythagorean theorem.

Applying the theorem we can write,

  $side=\sqrt{r^2+r^2}=\sqrt{2r^2}=r\sqrt{2}$

Therefore area of the square is,

  $Area=r\sqrt{2}\times r\sqrt{2}=2r^2$

Now area of the shagged region is

  $\text{Area of the shaded region}=\pi r^2-2r^2=(\pi -2)r^2$

Now according to the given inequality,

  $\begin{array}{l}(\pi -2)r^2\ge \text{9(}\pi -2\text{)}\\ \Rightarrow r^2\ge \text{9}\\ \Rightarrow r\ge \sqrt{9}\\ \Rightarrow r\ge 3\end{array}$

Therefore for $r\ge 3$ the area of the shaded region of the given figure will be greater than or equal to $\text{9(}\pi -2\text{)}$ .