Gaussian Technologies Department of Mathematics Now, how about this needle problem? obviously it seems hard to visualize what is the total possible amount of outcomes to help us, we create a coordinate system. Define x and 6 as shown below where x is the distance OP from the midpoint of the needle to the nearest crack, and 0 is the smallest angle between op and the needle Notice that a random toss of the needle can be 'marked out by this new coordinate system with the variables in the interval 0≤x≤1and0≤≤ which covers all the random positions the needle can fall to. Further, notice that the outcome with are interest in can be written as the inequality x<cos e By plotting the graph x= cos 0 as follows MISCELLANEOUS CALCULUS RELATED PUZZLESGaussian Technologies Department of Mathematics MISCELLANEOUS > CALCULUS RELATED PUZZLES Now, how about this needle problem? Obviously, it seems hard to visualize what is the total possible amount of outcomes. To help us, we create a coordinate system. Define x and ! as shown below where x is the distance OP from the midpoint of the needle to the nearest crack, and ! is the smallest angle between OP and the needle. Notice that a random toss of the needle can be ‘marked’ out by this new coordinate system with the variables in the interval 0 ! x ! 1 and 0 ! " ! # 2 which covers all the random positions the needle can fall to. Further, notice that the outcome with are interest in can be written as the inequality x < cos! By plotting the graph x = cos! as follows, P x cos! ! 1 O