Why are manholes covers always circular disks and not squares plates, though squares seem to meet the symmetry of the roads better than circles?
A very simple yet thoughtful application of geometry in our day to day lives. A circular manhole cannot fall through its opening, because for that to happen the size of the opening should be atleast as big as the diameter(longest chord) of the circle. But in case of square, it can easily fall through the opening if titled diagonally, as the side of the square is smaller the diagonal of the square ( diagonal = 1.414 times the side)
If two functions, both defined over the entire real number line are not derivable for a given value of x = c, can the sum of the functions be derivable at x = c?
Now lets first talk about, why a function is not derivable in the first place? it can be because of the following reasons:- 1) the function is not defined for x = c (ruled out for this question) 2) function is defined at x = c, but is not continuous 3) function is continuous, but not smooth at x = c Lets consider case 2:- f(x) = 1 for x<0 and f(x) = 0 for x >= 0 g(x) = -1 for x<0 and f(x) = 0 for x>= 0 now both f(x) and g(x) are discontinuous and hence non derivable, at x = 0 but if we consider v(x) = f + g v(x) = 0 for all x, which is a derivable function.
For which statistical distribution, values of mean, median and mode equal?
Although one would like to approach this question with mathematical equations and a lot of calculations, one easy and very logical approach is :- For a statistical distribution to show such a property it has to have a symmetrical curve. Why? Because only then median is going to coincide with the mean. Now that we have mean equal to the median. Lets talk about the "mode". For mode to be equal to the median, the middle quantity should occur more frequently than others, i.e. probability of the median value should be the highest. All these conditions are satisfied only by the normal distrubution:- 1) symmetric (most important) 2) mean = median = mode