Question

In the diagram OCA and ODB are radii of a circle with centre O and radius  cm. Angle  adians. CD and AB are arcs of circles with centre O and radii  cm and  cm respectively. The  perimeter of the shaded region ABDC is 4.4r cm.

     i.       Find the value of .

   ii.       It is given that the area of the shaded region is 30 cm2. Find the value of .

Solution

i.
 

It is evident from the diagram that;

We are given that;

Now we find lengths of arcs AB and CD.

Expression for length of a circular arc with radius  and angle  rad is;

Therefore, for length of arc CD;

Hence;

Next, for length of arc AB;

Hence;

Finally;

   ii.
 

It is evident from the diagram that;

We are given that;

Now we find areas of sectors AOB and COD.

Expression for area of a circular sector with radius  and angle  rad is;

Therefore, for area of sector COD;

Hence;

Next, for area of sector AOB;

Hence;

Finally;

We have found in (i) that ;