1987 AJHSME Problems/Problem 22
Revision as of 17:31, 13 March 2009 by 5849206328x (talk | contribs) (New page: ==Problem== <math>\text{ABCD}</math> is a rectangle, <math>\text{D}</math> is the center of the circle, and <math>\text{B}</math> is on the circle. If <math>\text{AD}=4</math> and <math>...)
Problem
is a rectangle, is the center of the circle, and is on the circle. If and , then the area of the shaded region is between
Solution
The area of the shaded region is equal to the area of the quarter circle with the area of the rectangle taken away. The area of the rectangle is , so we just need the quarter circle.
Applying the Pythagorean theorem to , we have Since is a rectangle,
Clearly is a radius of the circle, so the area of the whole circle is and the area of the quarter circle is .
Finally, the shaded region is , so the answer is