1. Half the length of the base of the rectangle can be found as
24/sqrt 3
So....the whole base length = 48/sqrt 3 = 16sqrt 3
So.......Area of rectangle = 24 * 16sqrt 3 = 384sqrt (3)
Side of the square= 8sqrt (2) ....area of the square = S^2 = (8sqrt (2))^2 = 128
P( that a point falls into the square ) = 128 /( 384 sqrt (3) ) ≈ .192 = 19.2%
2. Area of eqquilateral triangle = (1/2) (16sqrt (3) )^2 * sqrt (3) / 2 = 192sqrt 93)
P( that a point falls into the equilateral triangle ) = 192sqrt (3) / 384sqrt (3) = 1/2 = 50%
3. Area of circle not including inside the square= pi *8^2 - 128
P that a point fals into this area = [ pi *8^2 - 128 ] / ( 384sqrt (3) ) ≈ .1098 = 10.98%
4. Probabilty that a point falls into 4. the parts of the rectangle that's not included in the triangle, circle, or square
The area of the equilateral trinagle takes up 50% of the area of the rectangle....so....the probability of falling outside the triangle, circle or square is also 50%