1.13: Rectangle with an inset circular arc

0606/12/O/N/20 — Question 11 · 9 marks

In this question all lengths are in centimetres and all angles are in radians.

1.13 diagram
The diagram shows the rectangle ADEFADEF, where AF=DE=rAF = DE = r. The points BB and CC lie on ADAD such that AB=CD=rAB = CD = r. The curve BCBC is an arc of the circle, centre OO, radius rr and has a length of 1.5r1.5r.
(a)
Show that the perimeter of the shaded region is (7.5+2sin⁡0.75)r(7.5 + 2 \sin 0.75)r.
[5]
(b)
Find the area of the shaded region, giving your answer in the form kr2kr^{2}, where kk is a constant correct to 22 decimal places.
[4]