Question

For what values of r will the area of the shaded region be greater than or equal to 9(π−2)? Write your answer as an inequality.

Answer 1

`r^2(\pi -2)\ge 9(\pi -2)\\ \\r\ge 3\ units`

Explanation:

Find the area of the shaded region in terms of r.

1. The area of the circle with radius r is

`A_1=\pi r^2\ un^2.`

2. The area of the square with diagonal 2r is

`A_2=(1)/(2)\cdot (2r)\cdot (2r)=2r^2\ un^2.`

3. The area of the shaded region is the difference

`A_(shaded)=A_1-A_2=\pi r^2-2r^2=(\pi -2)r^2\ un^2.`

Since the area of the shaded region must be greater or equal to
`9(\pi -2)\ un^2.,` then

`r^2(\pi -2)\ge 9(\pi -2)\\ \\r^2\ge 9\\ \\r\ge 3\ units`

This inequality is solved only in positive numbers, because r cannot be negative.