The dartboard in the diagram is composed of two concentric circles. The radius of the larger circle is twice as long as that of the smaller circle. The dartboard is further divided into four equal parts by two diameters of the larger circle. 200 darts are randomly thrown towards the dart board and 60% of them land off the dartboard. What is the expected number of darts that land in the blue region?
The area of the smaller circle = pi *r^2
The area of the dartboard itself is pi (r + r)^2 = pi (2r)^2 = 4 pi * r^2
So the area between the circles is 4pir^2 - pi*r^2 = 3 pi*r^2
And the blue area is 1/4 of this = (3/4) pi * r^2
So.....the ratio of the blue area to the total area of the dartboard is (3/4) / 4 = 3/16
If 60% of the darts miss the board, then 40% must hit the board = 200 * .40 = 80
And (3/16) of these will land in the blue area = (3/16) (80) = 3 (80/16) = 3 * 5 = 15 darts