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Twenty water tanks are decreasing in size in such a way that the volume of each tank is $\frac{1}{2}$ the volume of the previous tank. The first tank is empty, but the other 19 tanks are full of water.

Would it possible for the first water tank to hold all the water from the other 19 tanks? Motivate your answer.

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Let $V$ be the volume of the first tank.
$\dfrac{V}{2} ; \frac{V}{4} ; \dfrac{V}{8} \ldots$
$S_{19}=\dfrac{\dfrac{V}{2}\left[1-\left(\dfrac{1}{2}\right)^{19}\right]}{1-\dfrac{1}{2}}$
$=\dfrac{524287}{524288} V$
$=0,9999980927 V$
$<V$
Yes, the water will fill the tank without spilling

OR

Let $V$ be the volume of the first tank.
$\dfrac{V}{2} ; \dfrac{V}{4} ; \dfrac{V}{8} \ldots$
$S_{19}=\dfrac{\dfrac{V}{2}\left[1-\left(\dfrac{1}{2}\right)^{19}\right]}{1-\dfrac{1}{2}}$
$=V\left[1-\left(\frac{1}{2}\right)^{19}\right]$
$<V .1$
$<V$
Yes, the water will fill the tank without spilling

OR

Let $V$ be the volume of the first tank.
$\dfrac{V}{2} ; \dfrac{V}{4} ; \dfrac{V}{8} \ldots$
$S_{\infty}=\dfrac{\dfrac{V}{2}}{1-\dfrac{1}{2}}$
$=V$

Since the first tank will hold the water from infinitely many tanks without spilling over, certainly:

Yes, the first tank will hold the water from the other 19 tanks without spilling water.

OR
If the tanks are emptied one by one, starting from the second, each tank will fill only half the remaining space, so the first tank can hold all the water from the other 19 tanks.

by Diamond (75,918 points)

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