E.L.E. - Imminent impact - i.e. the area of a spherical cap
What is the probability that a meteorite falling on the surface of the Earth, hit a certain area? This applet simulates meteorite impacts with the Earth's surface idealized as a sphere. The impacts are distributed evenly over the surface, and the frequency of those which occur within a given spherical cap is calculated.
You can set the position and height of the cap in the northern hemisphere respectively moving the point and the slider . You can define the number of impact points. Increase the number of impacts in order to find a good balance between the accuracy of computation of the probability of impact and the time of computation, then measure the relation between the height of the cap and the probability: for any value of that you choose, enter a point having as abscissa and the frequency as ordinate (do not forget the point ). What kind of relation do you get? Use the Best Fit Line tool or the FitLine command and measure the slope of the line you get: taking into consideration the fact that the radius of the sphere used is 1, can you derive a formula for the area of a spherical cap of height in a sphere of radius ?