Let ${\text{OP}} = x$.

(i)     Find ${\text{PQ}}$, giving your answer in terms of $x$.

(ii)     Hence, write down an expression for the area of the rectangle, giving your answer in terms of $x$.

Find the rate of change of area when $x = 2$.

The area is decreasing for $a < x < b$. Find the value of $a$ and of $b$.

(i)     valid approach (may be seen on diagram)     (M1)

eg     ${\text{Q}}$ to $6$ is $x$

${\text{PQ}} = 6 - 2x$     A1     N2

(ii)     $A = (6 - 2x)\sqrt {6x - {x^2}}$     A1     N1

[3 marks]

recognising $\frac{{{\text{d}}A}}{{{\text{d}}x}}$ at $x = 2$ needed (must be the derivative of area)     (M1)

$\frac{{{\text{d}}A}}{{{\text{d}}x}} =  - \frac{{7\sqrt 2 }}{2},{\text{ }} - 4.95$     A1     N2

[2 marks]

$a = 0.879{\text{  }}b = 3$     A1A1     N2

[4 marks]