16.
Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{858e6727-8498-462a-8064-d65254f1fd0f-08_478_938_283_502}
Figure 1 shows a sketch of the cardioid \(C\) with equation \(r = a ( 1 + \cos \theta ) , - \pi < \theta \leq \pi\). Also shown are the tangents to \(C\) that are parallel and perpendicular to the initial line. These tangents form a rectangle \(W X Y Z\).
- Find the area of the finite region, shaded in Fig. 1, bounded by the curve \(C\).
- Find the polar coordinates of the points \(A\) and \(B\) where \(W Z\) touches the curve \(C\).
- Hence find the length of \(W X\).
Given that the length of \(W Z\) is \(\frac { 3 \sqrt { 3 } a } { 2 }\),
- find the area of the rectangle \(W X Y Z\).
(1)
A heart-shape is modelled by the cardioid \(C\), where \(a = 10 \mathrm {~cm}\). The heart shape is cut from the rectangular card \(W X Y Z\), shown in Fig. 1. - Find a numerical value for the area of card wasted in making this heart shape.
(2)
[0pt]
[P4 January 2003 Qn 8]