My grandfather (an engineer who specialized in pumps) used to call these kinds of problems "the culling of true engineers from mathematical theorists." One story I remember him telling....
Take a rubber ball with a diameter of 1 meter. Drop it from a height of 10 meters, as measured from the center of the ball. It bounces, and ascends to a height of 9 meters. It bounces again and ascends to a height of 8.1 meters. From this we conclude that, with each bounce, the ball will ascend to 90% of its previous height. How many times will the ball bounce before it is at rest?
A mathematician will say that the ball never comes to rest; like the limit of 1/
x -> 0, it will approach a state of rest but never reach it.
An engineer will say that once the "bounce" is less than the radius of the ball (again, we are measuring from the center), the ball will exhibit no perceptible bounce and be at rest for all intents and purposes. If you wanted to turn this into a trick question, maybe...
The following function defines the height of each bounce of a rubber ball, as measured from the ball's center. Find the minimum x such that the ball is at rest.
f(x) = f(x - 1)
f(0) = 10
The domain of x is positive integers
The first instinct of a mathematician will be to take the limit of an infinite sum :evilgrin:
The answer, by the way, is 22, as f(22) is the first case where the result is less than 1.
AddedOr if you really want to be nasty...
A is the ordered set of all integers.
B is the ordered set of all even integers.
Which answer is true?
1) A > B because A contains every element in B and A contains elements that are not in B.
2) A < B because every element B(x) = 2A(x).
3) A = B because both are infinite sets.
4) All of the above.
5) None of the above.
I believe the correct answer is (5): there is no definition of equality between infinite sets.
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