Minus' Solution to

I'm Quakers for you!
Posted March 17 - 23, 1997

When the distance each of the ducks travels is put together, they must travel the distance of the diameter of the lake, which is 200 meters.

When Dahlia had been gliding for 75 minutes, she had progressed, 76 meters. When Danny had been gliding for 75 minutes, he had progressed 100 meters. This is a total of 176 meters, so the actual time is greater.

If Dahlia had been gliding for 90 minutes, she had progressed 90 meters. If Danny had been gliding for 90 minutes, he had progressed 120 meters. This is a total of 210 meters, so they would have passed each other, so the actual time is less.

After 86 minutes, Dahlia will have travelled 86 meters.
After 84 minutes, Danny will have travelled 112 meters. He rests until 84.5 minutes, and then travels 2 more meters to get up to 85.5 minutes. He rests for half a minute, taking him to 86 minutes, which means he has travelled 114 meters in 86 minutes.

86 + 114 = 200, so Dahlia and Danny will meet after 86 minutes.

Your Solutions

Mr. T and his Little Einsteins in Texas are back from their Spring Break to feed our grouchy mascot. Minus has been trying to get a hold of you guys, but your e-mail doesn't seem to be working. So if you're reading this message, Mr. T, get in touch with your toothy friend!

At first glance, this appears to be your basic D=RT problem. However, after further review, it became evident that we were lambs on the quick path to the slaughterhouse.

To determine Dahlia's and Danny's respective rates, you must combine the swimming rates with the rest periods and we get 1 meter/min and (4/3) meter/min respectively.

Now comes the twist. If we assume that these rates are constant throughout their journeys, we would use the equation:
200 = 1T + 4/3T
200 = 7/3T
T = 85 5/7 minutes or approximately 86 minutes.

Tragically enough, our assumption would have proved fatal and we would have gotten the Big X.

The reason for this incorrect assumption is that our rates are based on each of the ducks resting after every 1 minute burst of energy. That is fine up until the seconds surrounding their romantic rendevous. We would not need to account for either of them resting, therefore, throwing off the assumed average speed.

The situation is easily rectified. Since the greatest common multiple of 1 & 7 less than 200 is 196, we know that:
196 = 1T + 4/3T
196 = 7/3T
T = 84

Meaning that after 84 minutes our two lovebirds are exactly 4 meters apart and fully rested. Therefore, they each must travel only 2 more meters in the next minute to bring them together in precisely 85 minutes. Checking our answer, we determined that since Dahlia's average rate for the 1st 84 minutes was 1 meter/min., she travelled 84 meters plus 2 meters in the last minute for a total of 86 meters. Likewise, Danny travelled 112 meters in the 1st 84 minutes ((4/3)(84)) plus his final 2 meters for a total of 114 meters.
86 + 114 = 200 m

I always suspected the man was doing well over 50% of the work to develop relationships. Now we have proof positive!!!


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