Sprinklers are crop irrigation equipment which rotate around a center and spay water on the crops in the circular region. Two such high power sprinklers, occupying negligible area are installed in a straight line in a field such that they spray water on a common area. Shown below are the side and top views where points A and B are the sprinklers.

Both the sprinklers spray over an equal area. It is given that, $\mathrm{CD} = 400 \mathrm{m}$ and $\angle \mathrm{CAD} = \angle \mathrm{CBD} = 90°.$

i) Find the radius of the circular region sprayed by the sprinkler.
ii) Find the area of the overlapping region.
 

$AB$ is the diameter of a circle, centre $O, C$ is a point on the circumference such that $\angle COB=\theta$. The area of the minor segment cut off by $AC$ is equal to twice the area of the sector $BOC$. Prove that  $\sin \frac{\theta }{2}\cos \frac{\theta }{2}$$=\pi$ $\left(\frac{1}{2}-\frac{\theta }{120}\right)$.

In the given figure, $O$ is the centre and $AOB$ is the diameter of the circle. Find the area of the shaded region. 

A round table cover has six equal designs as shown in figure. If the radius of the cover is $28 \mathrm{c}\mathrm{m},$find the cost of making the designs at the rate of $\mathrm{R}\mathrm{s}.0.35$ per ${\mathrm{c}\mathrm{m}}^2.$ (Use $\sqrt{3}=1.7$)

A horse is tied to a peg at one corner of a square shaped grass field of side $15 \mathrm{m}$ by means of a $5 \mathrm{m}$ long as given in figure. Find the increase in the grazing area if the rope were $10 \mathrm{m}$ long instead of $5\mathrm{}\mathrm{m}.$ (use $\pi =\mathrm{}3.14$)

A circle has radius 5 cm. Three chords of lengths 6 cm, 8 cm and 10 cm are drawn.

Which chord subtends the largest angle on its minor arc?

Raghu was playing in his grandfather's old garage, where his grandfather kept many antique items. In a dark corner, he found a really old clock. He found that the clock was dead at $9:05 \mathrm{a}.\mathrm{m}$ . He pulled a lever on the clock and the minute hand shifted to $5$.

If the length of the minute hand of a clock is $3.5 \mathrm{cm}$, find the area swept on the face of the clock by it between $9:05 \mathrm{a}.\mathrm{m}.$ and $9:25 \mathrm{a}.\mathrm{m}.$ (Use the value $\mathrm{\pi }=\frac{22}{7}$)

In the figure, if $ABC$ is an equilateral triangle, then shaded area is equal to: