Problem: a a b b
Answer Choices:
A. a b b a β β
B. a b b a β
C. a 2 b 2 b 2 a 2 β
D. a + b 2 b 2 b a + b β
E. a 2 + 2 a b b 2 + 2 a b b 2 + 2 a b a 2 + 2 a b β
Solution:
The area of the shaded region is
Ο 2 ( ( a + b 2 ) 2 + ( a 2 ) 2 β ( b 2 ) 2 ) = Ο 2 a + b 2 ( a + b 2 + a β b 2 ) = Ο ( a + b ) a 4 2 Ο β ( ( 2 a + b β ) 2 + ( 2 a β ) 2 β ( 2 b β ) 2 ) = 2 Ο β 2 a + b β ( 2 a + b β + 2 a β b β ) = 4 Ο ( a + b ) a β
and the area of the unshaded region is
Ο 2 ( ( a + b 2 ) 2 β ( a 2 ) 2 + ( b 2 ) 2 ) = Ο 2 a + b 2 ( a + b 2 + b β a 2 ) = Ο ( a + b ) b 4 2 Ο β ( ( 2 a + b β ) 2 β ( 2 a β ) 2 + ( 2 b β ) 2 ) = 2 Ο β 2 a + b β ( 2 a + b β + 2 b β a β ) = 4 Ο ( a + b ) b β
Their ratio is a / b a / b
