Solving the Interesting Problem of the Last Bit of Water in a Bottle

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In summary, there is a popular myth that the last bit of water in a bottle is mostly backwash. However, after trying to test this myth by using variables x and y to represent the sips and backwash, and applying a differential equation (dy/dx = 0.05 - 20(y/(1000-(20-0.05)x))), the person is unable to solve it. They suggest a change of variables and provide a resource for solving the problem.
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There is a popular myth that after drinking a bottle of water, the last bit is mostly backwash. Well I decided to try and test it, but got stumped.

Lets call y the amount of backwash in the bottle
Lets call x the number of sips taken
The volume of the bottle will be 1000mL
Assume each sip is 20mL
Assume that each sip backwashes 0.05mL into the bottle

dy/dx= 0.05 -20( y / (1000 -(20-0.05) x ) )

I can't separate variables here, so I do not know what to do. This is not a homework problem, I was just wondering if anyone could help me solve this differential equation. At 51 sips there will be nothing left in the bottle.
 
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I tinkered with it a bit... try making the change of variables [itex]x' = 1000 - (20 - 0.05)x[/itex], then look at http://en.wikibooks.org/wiki/Differential_Equations/Exact_1 . I didn't take the calculation all the way through but it looks solvable that way.
 
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First of all, it's great that you're trying to test this popular myth! It's always important to question and experiment with things we hear or assume to be true. Now, let's dive into solving this interesting problem of the last bit of water in a bottle.

From the given information, we can set up the following differential equation: dy/dx = 0.05 - 20(y/(1000 - (20 - 0.05)x)). This equation represents the rate of change of backwash in the bottle with respect to the number of sips taken.

To solve this equation, we can use the method of separation of variables. First, we can multiply both sides by (1000 - (20 - 0.05)x) to get rid of the denominator on the right side. This gives us: (1000 - (20 - 0.05)x) dy/dx = 0.05(1000 - (20 - 0.05)x) - 20y.

Next, we can integrate both sides with respect to x. This gives us: ∫(1000 - (20 - 0.05)x) dy = ∫(0.05(1000 - (20 - 0.05)x) - 20y) dx.

The left side integrates to 1000y - (10 - 0.025)x^2 + C, where C is the constant of integration. The right side integrates to 50x - 0.025x^2 + C. Thus, our equation becomes: 1000y - (10 - 0.025)x^2 + C = 50x - 0.025x^2 + C.

Now, we can solve for y in terms of x: y = (50x - 0.025x^2 + C + (10 - 0.025)x^2 - C)/1000. Simplifying this, we get: y = (50x - 0.025x^2 + 10x^2)/1000. This can be further simplified to y = (50x + 9.975x^2)/1000.

Now, to find the amount of backwash left in the bottle after 51 sips, we can plug in x = 51 into our equation. This gives us: y = (50(51) + 9.975(
 

Related to Solving the Interesting Problem of the Last Bit of Water in a Bottle

1. What causes the last bit of water to remain in a bottle?

The last bit of water in a bottle is typically caused by surface tension. As the water level decreases, the surface tension between the water and the bottle increases, making it difficult for the water to flow out of the bottle.

2. How can I easily remove the last bit of water from a bottle?

One method is to simply tilt the bottle at an angle and pour the water out slowly. This will reduce the surface tension and allow the water to flow more easily. Another option is to use a straw or pipette to suck out the remaining water.

3. Why is the last bit of water important to remove?

The last bit of water in a bottle may contain impurities or bacteria that can make the water unsafe to drink. It is important to remove this water to ensure the safety and cleanliness of the remaining water in the bottle.

4. Are there any other methods for removing the last bit of water from a bottle?

Yes, there are several other methods that can be used to remove the last bit of water from a bottle. These include using a paper towel or cloth to absorb the water, using a hair dryer to evaporate the water, or using a funnel to transfer the water to another container.

5. Is there any way to prevent the last bit of water from remaining in a bottle?

There are a few ways to prevent the last bit of water from remaining in a bottle. One option is to use a bottle with a wider opening, which will reduce the surface tension and make it easier for the water to flow out. Another option is to use a bottle with a spout or nozzle, which will also make it easier to pour out the water. Additionally, making sure to thoroughly dry the bottle after each use can help prevent the build-up of surface tension.

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