Someone in my household put a long metal water bottle in the dishwasher. Let's simplify and assume that this water bottle is just a 10" long and 2" diameter cylinder. Disregard that the top part is narrower than the rest.

I am wondering what are the chances that throughout the dishwasher cycle at least one water drop will reach bottle's bottom from the inside? Assume that we had a sweet lemonade in the bottle before and really need this stickiness washed. Any thoughts are accepted. We don't exactly know from where dishwasher shoots water, so assume random directions.

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## Friday, May 28, 2010

### Metal Water Bottlle in the Dishwasher?

Posted by
Maria
at
10:12 AM

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Labels:
hard,
mother-in-law

## 8 comments:

I think we are given not quite enough information. Other factors will include time, and of course the interior diameter of the bottle even tho the problem asks only about the BOTTOM of the bottle. Might want to know the length (height) of the 2" neck and its ratio to the 10" bottle Then we'd ask to know the intensity of the spray, the speed of the water drops or streams. Let's see, there's also likely to be splashing inside the bottle....

But even given all that information I think I can only roughly guess that the solution would involve something like rays of light scattering from a half-sphere. But there's still gravity. Gosh I'm licked.

I know that to answer precisely we need to know more about what happens inside the dishwasher. But I feel that even with what we have we can come up with some interesting speculations. I added a picture of a water bottle and an assumption that it is a perfect cylinder. I think we can assume that water shots are coming from all the possible directions. How many of them will get inside? How many will reach the bottom?

I am pretty sure the layer on the inside of the majority of those style bottles means they are not dishwasher safe. I think they are meant to be washed out after each use so nothing sticks to the bottom of it. Sorry, I know this isn't helpful here but I am pretty sure the dishwasher is really bad for the inside of these bottles. Good luck.

I am doing a lot of speculating here. You said drops are coming from all directions. That would also mean (to me at least) that dops would come from different distances. For instance, one place drops may come from would be 2" from underneath the lip of the bottle. That would be 360 degrees in all directions in just one place and 360 degrees in each degree of that plane and each degree can be subdivided into infinitesimal sectors. Multiply that by different distances, different angles and different places outside the bottle and the possibilities are endless. I hope this sounds logical.

However, let's assume it would be where most people would put a water bottle. I would put it near the center over the sprayer.and sprinkles would come from most directions. I imagined cutting the bottle in half lengthwise. The sides would be 10" and the top and bottom would be 2" each. The perimeter is 24". If the water is shooting in a relatively straight line the drops would only hit the bottom 1/12th of the time. However, I think it would actually be less than this because of the greater exposure of the sides especially to multiple angles.

That's "drops", not "dops".

I agree that such bottles should not be washed in the dishwasher. It is so easy to wash them by hand. If one wants to disinfect, I think boling water and soap should be enough. Even if they happen to be dishwasher safe, I am pretty sure it is pointless. It seems to me that chances of water reaching the bottom should be close to zero, unless you specially position the bottle atop the spray. But even then it will probably be pushed away by the spray at some point.

Gurubandhu offers interesting thoughts and gets a puzzle point for daring. He suggests that from all the water sprays that do enter the bottle opening, only 1/12 or less reach its bottom. The rest will spray the walls from the inside. I would say it is probably correct. I made a sketch here and saw that if water sprays right from the center of the opening it will reach the bottom in approx 12 degree angle out of all 180 degrees. But what about cases when water doesn't even enter the bottle and just gently touches it from outside? Anyone dares to put her/his thoughts to it? There is still a wealth of puzzle points to earn. And of course we all want to know and warn our mothers-in-law to never ever do it again.

Actually, I think it would be 1/24 since You would have to multiply the perimeter by 2 since you have to consider both the inside and outside surfaces. The sprays from the outside of the bottle cannot reach to opposite side of the bottle on any side except from the open bottom.

So, you think it is around 1/24 or 4%...

My calculations are closer to 6%. Of course this all is just speculations as we know that we shouldn't be washing this bottle in the dishwasher to start with, bottle could be positioned anywhere in the dishwasher and may get lucky and be right on top of the spray, or completely blocked from it, dishwashers have different mechanisms. Here is my drawing for an average situation. What do you think?

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