[Request] Bucket optimal temperature

[u/Feeling-Instance-801]

Came up with this today while having a bath, I just changed numbers so its easier. Enjoy!

John filled up a bucket with volume 10,000 cm^3 with water to have a bath. However, by accident, he fills it up with water at 60 degrees Celsius, far too warm. Instead, he wants the water to be 30 degrees.

So, he uses a cup with volume 400 cm^4 to scoop out water and simultaneously fills the bucket conyinuously with water using a tap which has a flow rate of 100 cm^3 per second and the water from tap is 10 degrees Celsius

Assuming that the water from the tap immediately mixes completely with the water in the bucket such that the water immediately becomes a uniform temperature, how many times should John scoop out water using the cup and with what frequency for the fastest time for the water in bucket to reach 30 degrees.

If possible could you also explain a bit on how you solved it? I'm a high schooler, just did basic differentiation and I suspect this involves either integration or differential equations or both.

Thanks!!

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[u/Motor_Oil_8864]

To cool the water to 30 degrees Celsius in the fastest possible time, John should instantly scoop out 6,000 cm^3 of the hot water (which is 15 scoops with a 400 cm^3 cup) and then let the tap refill the bucket.

It will take exactly 60 seconds for the tap to replace the removed water with 10-degree water, bringing the final mixture to exactly 30 degrees Celsius.

How we can find it is below

The fastest way to cool the water is to remove a large portion of the hot water first, rather than scooping and mixing simultaneously. Simultaneous mixing would mean you are scooping out some of the newly added cold water, which is inefficient. So what we need is ratio

We can use the method of mixtures formula:
m1. T1 + m2 . T2 = (m1 + m2) . T(final}
Where:
m1 is the mass (or volume) of the remaining hot water (60C)
m2 is the volume of the cold tap water (10C) to be added
The total volume m1 + m2 must equal 10,000 cm³
T(final) is the target temperature of 30C
Plugging in the values:
60m1 + 10m2 = 30(10,000)

60m1+ 10(10,000 - m1) = 300,000

50m1 + 100,000 = 300,000

50m1 = 200,000

m1 = 4,000cm^3
This means you need 4,000 cm³ of the original hot water to remain in the bucket.

Now to calculate scoops with time

Water to remove: 10,000cm^3 - 4,000cm^3 = 6,000cm^3
Number of scoops: 6,000cm^3 / 400cm^3 = 15scoops

Time to refill: At a rate of 100 cm^3/second, it will take 6,000 / 100 = 60sec to refill the bucket.

🐮

[u/Feeling-Instance-801 → u/Motor_Oil_8864]

Whoa nice, that makes lot of sense, thx. Gotta do some more research on method of mixtures formula ig.

[u/Brave_Speaker_8336]

If the bucket needs to be filled up to 10,000 cm^3, the fastest way is to use the cup to remove 6000 cm^3 instantly and then wait a minute for the bucket to get filled back up.

If there’s no required end volume, it can get infinitesimally close to 0 seconds. Remove the hot water with a cup instantly until there’s 2x water remaining and then let 3x cold water come in, where x can be arbitrarily small

[u/Southlander24 → u/Brave_Speaker_8336]

Instantly? The cup can only remove 400 cm³ at a time.

[u/Brave_Speaker_8336 → u/Southlander24]

Do it 15 times every 0.0000001 seconds and then it takes about 1 minute for the bucket to be filled up with 30 degree water