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A spider and an ant walk along the circular edge of an open cylindrical tank, in opposite directions. They start walking from the same point and the spider walks at a speed of 3/4 rpm (revolutions per minute) while the ant at 2/3 rpm. Sometime between 22 and 23 minutes since they started walking, a
guard standing in the middle point of the tank holds a torch and shines with its beam the edge of the tank, aiming exactly at the point where the two insects had started walking. If the beam has an opening angle of 90o, can you find the probability that the guard sees both insects?
Dear bobbym,
I just fell on this very nice problem.
Can I ask something? I understand that you add this stack variable e to all 3 inequalities, to make them equations, right? Why does this variable has to be the same for all 3?
Thank you!
Hi phrontister;
Okay, thanks for coming in. See you later.
Hi chen.aavazi;
Calling x = apples and y = bananas, we can reduce the system down to:
A)
100 - x > 3 y
3 y > 4 x
4 x > 100 - y
We can now graphically solve for the answer, I used Geogebra.
http://i.imgur.com/4A8ZlQC.png
The answer lies in the small darker triangle and must be an integer for (x,y). That reduces down the possibilities to just a few, as a matter of fact there is just one integer coordinate in there.
We can go a bit further with a trick that is used in numerical work which I invented?!
The three inequalities in A) can be changed to equations with the addition of what I call a slack variable... This is a term used in linear optimization but I give it an added meaning.
If 100 - x > 3 y we only have to add something to the RHS to pick up the slack. I add e ( slack variable) which balances the inequality and gives us the equality of 100 - x = 3 y + e. We do that with all 3 inequalities and hope for the best.
100 - x = 3 y + e
3 y = 4 x + e
4 x = 100 - y + e
this can be solved by ordinary means
x = 19.0476, y = 26.1905, e = 2.38095
Remembering that x = apples and y equals bananas we can guess and try the closest integers and get apples = 19 and bananas = 26. We test to see that we are right and we are done!
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