Beauty of Mathematics

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry-- Bertrand Russel.
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Monday 30 January 2017

Questions on Time, Spped and Distance



Question1: A man leaves his house at H1:X  ( at a time X minutes past H1 hours) and returns home after a few hours at H2:Y ( at Y minutes past H2 hours). He finds that the position of the minutes hands and hour hands had interchanged during this period. In other words the angle between the hands was the same just when he left the house and when he just returned home. Find the value of X and Y in terms of H1 and H2.


Solution:   We know the speed of minute hand = 60 per minute, and
Speed of Hour hand = 300 per hour or ½0 per minute

Angle made by the Minute hand with vertical line ( 12-6 line in the clock) when he left the home at H1:X
Angle made by the Hour hand with vertical line when he returned home.

Therefore,                 6X = 30 H2+ Y/2……………. Equation 1

Similarly;
Angle made by the Hour hand with vertical line ( 12-6 line in the clock) when he left the home at H1:X
Angle made by the Minute hand with vertical line when he returned home.

Therefore,

30H1+X/2 = 6Y…………. Equation 2

Solving the above two equations we get

X= (720H2 +60H1)/143

Y= (720H1 +60H2)/143

From the above two values of X and Y, it can be found for how much  time was the man away from his home. Apart from the above, we can also derive a few more short cuts as mentioned below:

  

 The angle between the hands=(H2-H1)360/13

Time Away=(H2-H1)+(60/13)(H1-H2)