Your Practice Set – Applications and Interpretation for IBDP Mathematics
2 . The following diagram shows a circle with centre O , diameter AB and radius 10 . Let C be a point on the circumference with BOC ˆ
�
� � radians , where 0 ��
� . D is a point on
2 the x -axis such that CD and OD are perpendicular . A rectangle is inscribed in the circle . Let P be the total area of the shaded regions .
( a ) ( i ) Express CD and OD in terms of � .
( ii ) Express the area of the inscribed rectangle in terms of � .
( iii ) Hence , express P in terms of � and � .
( b ) ( i ) Find d P
d� .
[ 6 ]
( ii ) Solve the equation d P
0 d� � .
It is given that
2 d P d
1600sin�cos�
2
� � .
( iii ) Hence , find the value of � when P is at a local minimum , justifying that it is a minimum .
( iv ) Using ( a ) and ( b )( iii ) to find the minimum value of P .
( c ) Write down the two values of � for which P has its greatest value .
[ 8 ]
[ 2 ]
166
SE Production Limited