Question

the ratio off the volume of water in bottle P to the volume of water in bottle Q is 3:4 . Rajeev drank 40 ml of water from bottle P and the ratio then became 13:20 . how much water was there in bottle P at first??

Answer 1

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Say there is a water bottle that is filled with 300 mL of water and has a circular hole with a radius of 2 mm. In this bottle, the water sits 7.8cm above the top of the hole (which has been drilled 1.5cm above the bottom of the bottle).

According to Bernoulli's law the velocity v of the water flowing out is equal to 2gh−−−√

Therefore for the setup above, v=2∗9.81 m/s2∗0.078 m−−−−−−−−−−−−−−−−−−−√=1.24 m/s

Using this, the flow rate can be calculated as Q = Av = π(0.002 m)2∗1.24 m/s=0.000016 m3/s=16 mL/s

This doesn't seem accurate considering that the experimental flow rate is equal to 8 mL/s (40 mL over 5 seconds). However I understand that it ignores viscosity (and other things?)

I'm wondering a few things, firstly, does the theoretical math here apply to the situation I'm describing? The hole in the bottle isn't exactly a pipe and the only examples I've seen with water flow involve pipes.

Secondly, can Poiseuille's Law be used to determine the flow rate instead, with a more accurate result? (From what I understand Q=πPR^4/8nl, however I don't understand what P is, seeing as in Bernoulli's law pressure cancels and as aforementioned this isn't a typical pipe example.)

Thirdly, I assume the theoretical flow rate will still be different from the experimental flow rate, what factors cause this?

Answer 2

The ratio off the volume of water in bottle P to the volume of water in bottle Q is 3:4 . Rajeev drank 40 ml of water from bottle P and the ratio then became 13:20 . Bottle P at first has 300 mL of water.

Let us consider that the volume in Bottle P = p mL

And, the volume in Bottle Q = q mL.

Given, the ratio of volumes of water in Bottle P to Q is 3:4

Thus, we can write it as,

$\frac{p}{q}$ = $\frac{3}{4}$

⇒ 4p = 3q

⇒ p = $\frac{3}{4}$ q                                 ...(1)

If 40 mL of water from bottle P is drank, then remaining volume of water = (p - 40) mL

and the volume of water in bottle Q is the same = q mL

Given, this new ratio is 13:20

We can write this ratio as :

$\frac{p-40}{q}$ = $\frac{13}{20}$

⇒ 13q = 20p - 800

Replacing the value of p = (3/4)q in this equation, we get:

⇒ 13q = 20(3/4)q - 800

⇒ 13q = 15q - 800

⇒ 2q = 800

⇒ q = 400

As p = (3/4)q

So, p = (3/4) × 400 = 300

Thus, Bottle P has p = 300 mL of water and Bottle Q has q = 400 mL of water. So, Bottle P has 300 mL of water at first.