2020 NYJC P1 Q11
A farmer has a plot of the land, $OAB$, with adjacent sides $OA$ and $OB$. With reference to $O$ as the origin, $OA$ and $OB$ are parallel to the $x$-axis and $y$-axis respectively. The arc $AB$ is described by the parametric equations
$x=\cos \theta $, $y=\theta -\frac{1}{2}\sin 2\theta $, $0\le \theta \le \frac{\pi }{2}$.
(i)
Show that the area of the plot of land is $\frac{2}{3}$ units$^{2}$.
[4]
Due to a strong worldwide demand for latex, the farmer plans to use a portion of his land to grow rubber trees. He divides his land into three portions. One portion is a rectangle with vertices at $O$ and along arc $AB$, and sides parallel to $OA$ and $OB$. He intends to use only this rectangular plot of land which has an area of $K$ units$^{2}$ to plant rubber trees.
(ii)
By using differentiation, find the largest value of $K$.
[4]
In the midst of his planning, the futures contract for latex shot to historical high. To maximise his returns, the farmer decides to redistribute his plot of land to plant more rubber trees. He divides his plot of land into two portions using the line $y=2x+0.1$ and allocates the larger plot to plant rubber trees.
(iii)
Find the area of this larger plot of land meant for rubber trees.
[4]
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