A chord connecting two points randomly chosen on a circunference of a disk of radius 1 cuts it into two segments. Select the best approximation to the mathematical expectation of the area of the smaller segment.
1. O.2
2. 0.6
3. 0.4
4 0. 8
5. 1
Let the circle of centre O and radius 1 unit . A and B are two points on the circumference which cuts the circle into two segments. The portion under AB is a minor segment.
Area of smaller segment = Area of sector AOB - Area of Triangle AOB
Area of sector AOB for angle "\\theta" "= \\theta \\times \\dfrac{\\pi r^2}{360}"
Area of Triangle AOB "= \\dfrac{1}{2} \\times base \\times height."
"= \\dfrac{1}{2} \\times OA \\times BM"
"= \\dfrac{1}{2} \\times r \\times rsin\\theta = \\dfrac{1}{2}r^2sin\\theta"
Area of smaller segment "= \\dfrac{\\theta \\pi r^2}{360}- \\dfrac{1}{2}r^2sin \\theta"
Comments
Leave a comment