2021 NYJC Promo Q11

A designer wishes to create a piece of artwork with painted area of \(1200\) cm\(^{2}\) on a rectangular piece of canvas. The painted area measures \(x\) cm by \(y\) cm and is surrounded by an unpainted border with top and bottom margins of \(3\) cm each, and side margins of \(4\) cm each on the canvas, as shown in the diagram below.

1. By differentiation, find the dimensions of the canvas with the smallest area.[6]
2. What is the largest possible area of the canvas if \(30\le x\le 50\)?[2]

At an exhibition, a spotlight illuminates a circular region of radius \(\frac{2}{\sqrt{\pi }}\) cm on the artwork. The area of this circular region then increases at a constant rate of \(20\) cm\(^{2}\) per minute.

3. Find the rate of change of the radius after \(3\) minutes.[4]