The time between seeing a lightning flash and hearing the resulting thunder gives an accurate estimate of the distance to its location. During one storm a meteorologist observes 29 such flashes, and calculates the distances in miles as follows:
6.02,
3.01,
0.69,
3.38,
2.01,
4.69, 3.54,
3.67,
3.67,
4.55,
5.79,
3.72, 1.05,
3.73,
3.98,
7.28,
2.10,
2.90, 6.95,
6.18,
7.20,
4.89,
6.60,
2.53, 2.09,
1.30,
1.81,
4.75,
6.91
Assume that the cloud is circular, is not moving, and that the lightning strikes are uniformly distributed through the cloud. Determine the diameter of the cloud.
The cloud has a diameter of about 7 miles.
(If I am wrong, should lightening strike me dead.)
If you read the previous comment, you will see that I outlined a formal solution method (which I chose to not use):
Make a probability function of a strike distance as a function of the variables: the cloud's ground distance (d), its diameter (2r), and its height (z). This is just the distribution of the distances of the points in such a disk to the ground observer and can be found from geometry. The expected distances make a histogram (for 29 strikes) that may be compared with the data. The shape of the expected histogram is controlled by d, r and z.
The RMS between the expected and observed histogram is partially differentiated w.r.t. d, r, & z, set to zero, and the formal best fit values to these variables is found.
I did it the easier way: I simulated the situation, made a grid of solutions, using approximately 50x50x50 clouds, had each experience 29000 random strikes, and found the minimal RMS to the data in each case.
I found the best fit was for a cloud center about 4 miles away as measured along the ground, at a height of 0.1 to 0.5 miles (yes - a big spread) and a diameter of 6.9-7 miles (so you see it is largely overhead, and thus allowing the close strikes).
The data were never very well fit, especially the high number: six strikes between 3.5 and 4 miles and the dearth of strikes between 5 and 5.5 miles. More data would yield higher confidence results.
A selection of the best fits (those fits nearest the minimum Chi-square) are
here and the program is
here.
I would make a nice 3-D plot of the Chi-sq space, but I don't know a good Mac app for this. Anyone?
Edited on February 9, 2019, 9:53 pm
solution
