inequalities help

[merry cedar]

<@&791435371564892232>

[rigid ridge]

The string is cut into 2 circles and they happen to be the circumferences of each circle. The total length is 18π.
As they are circumferences, they will be equal to 2πr in relation to the radius.
First radius is r, so the length of string used for that circle is 2πr. The length of string used for the other circle is 18π - 2πr, which can be simplified to 2π(9 - r).
That is in the form 2πr, but r is actually (9-r). Therefore, the radius of the second circle is (9 - r)

With the general area of a circle being πr², you can make the total area for the circles > 53π.
53π < πr² + π(9 - r)²
53 < r² + (9 - r)²
53 < r² + 81 - 18r + r²
0 < 2r² - 18r + 28
0 < r² - 9r + 14

The other part of that can be done by inspection, or putting the whole thing into the quadratic equation, or completing the square, the basic quadratic factorisation methods

@merry cedar