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Originally Posted by Ghoulish Delight
Will Achilles ever catch up?
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Yes he will. At a hare* over 1,111.111 meters. I had to brush up on my algebra to do this. Okay, I cheated and found a web site to help me solve. But I DID come up with the formula on my own. Formula wasn't perfect, but it did help me find the correct answer (cuz I knew that Achilles would catch the tortoise).
My thinking was that x would equal the point where Achilles would catch the tortoise (the break-even point if you will). so I came up with the formula:
10x = x + 1000
And then I took this formula to an
Algebra web site and plugged my formula in. It gave me this:
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Simplifying
10x = x + 1000
Reorder the terms:
10x = 1000 + x
Solving
10x = 1000 + x
Solving for variable 'x'.
Move all terms containing x to the left, all other terms to the right.
Add '-1x' to each side of the equation.
10x + -1x = 1000 + x + -1x
Combine like terms: 10x + -1x = 9x
9x = 1000 + x + -1x
Combine like terms: x + -1x = 0
9x = 1000 + 0
9x = 1000
Divide each side by '9'.
x = 111.1111111
Simplifying
x = 111.1111111
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I knew that the answer was not 111.1111+ meters. (presumably the "1"s after the decimal point repeat ad infinitum, but I don't know how to put the bar over it like I could when I was handwriting, so just imagine it is there.) But logic dictated that it would be 1,111.1111 meters (or 111.1111 meters past the effective start point for the tortoise when Achilles started). Granted, I would still get a failing grade in algebra, but I knew the answer was correct.
Basic logic also dictates that Achilles would of course catch the tortoise. The relationship in their speeds is not exponential, it is linear. Both speeds are (presumably) constant, so of course Achilles would catch the tortoise.
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Well, there's a half hour of my life I'll never get back. :geek: