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In a public square, there is a fountain that is formed by two cylinders, one with radius $r$ and height $h_1$, and the other with radius $R$ and height $h_2$. The middle cylinder fills and, after overflowing, starts to fill the other one.

enter image description here

If $R= r \sqrt{2}$ and $h_2=\frac{h_1}{3}$ and, to fill the middle cylinder, it took $30$ minutes, then, to fill this source and the second cylinder so that it is completely full, how many minutes will it take?

I made the following attempt:

If the middle cylinder has a volume equal to $\pi r^2 3h_2$, and the second cylinder has a volume equal to $\pi \left(r \sqrt{2}\right)^2 h_2$, then we have $x$ are the minutes it takes to fill the font completely: $$\frac{\pi r^2 3h_2}{\left(\pi \left(r \sqrt{2}\right)^2 h_2\right)+\left(\pi r^2 3h_2\right)}=\frac{30}{x} $$ $$x= 50$$

But the answer is 40 minutes. Where am I going wrong?

Lambert macuse
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    If you have a fountain, shut it up; let the fountain rest. (K. Prutkov) – markvs Nov 24 '21 at 20:36
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    Given how you're using the variables later, I think you mean $h_2=\frac{h_1}{3}$, but that shouldn't affect your answer. What does affect your answer, though, is that the middle portion of the wider cylinder starts out already filled (by the central cylinder). – Robert Shore Nov 24 '21 at 20:37

1 Answers1

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Note that part of the second cylinder includes the middle cylinder so by the time that the middle cylinder is filled, you have already filled a portion of the second cylinder. The total volume of the fountain is $(\pi(r\sqrt{2})^2h_2)+(\pi r^2(3h_2))-(\pi r^2 h_2)=4\pi r^2h_2$

Hence, we have $\frac{3\pi r^2h_2}{4\pi r^2h_2}=\frac{30}{x}\implies x=40$

Alan Abraham
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  • Why did you subtract by $\left(\pi r^2 h_2\right)$? – Lambert macuse Nov 24 '21 at 21:21
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    Because that's the volume of the intersection of the two cylinders. By the time the middle cylinder has been filled and starts overflowing, that intersection has already been filled and doesn't need to be filled again.

    The easy way to see that the answer is $40$ minutes, given what you've already done, is that it takes $30$ minutes to fill the central cylinder so it took $10$ minutes to fill that cylinder to the height of the wider cylinder. Thus, you can subtract $10$ minutes from your calculation to get the answer.

     – Robert Shore Nov 24 '21 at 21:30