NYJC/2021/JC1/Promo/P1/Q11

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

(i) By differentiation, find the dimensions of the canvas with the smallest area.

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(ii) What is the largest possible area of the canvas if $30 \leq x \leq 50$ ?

[2]

At an exhibition, a spotlight illuminates a circular region of radius $\frac{2}{\sqrt{\pi}} \mathrm{cm}$ on the artwork. The area of this circular region then increases at a constant rate of $20 \mathrm{~cm}^2$ per minute.
(iii) Find the rate of change of the radius after 3 minutes.

[4]