Quadratic Functions, Equations and Inequalities

A rectangle has perimeter \(24\) cm and width \(x\) cm.

1. Show that its area, \(A\text{ cm}^2\), can be expressed as

   \[ A=36-(x-6)^2. \]

   State the practical range of \(x\).

2. Find the greatest possible area and the corresponding dimensions. Explain why an area of \(40\text{ cm}^2\) is impossible.

3. Find the dimensions when the area is \(32\text{ cm}^2\). Explain why the two algebraic roots describe the same rectangle.