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Dude, say what you've written out loud and tell me if it makes sense to you. It doesn't, to me.If the total pressure at the top caused the water to move up, water would be sprouting out from the straw even without the child applying any pressure on it. And with greater intensity! - since without the child sucking on the straw the pressure would be even greater at 1.0p₀!
The following is my explanation:
So p₀ is the pressure inside the straw including where it is touching the liquid. Now, since the kid is only able to produce a pressure difference of 0.1p₀, we need to find out how much difference this change in pressure will produce in the height of the liquid (inside the straw, ofcourse).
Δp=Δhρg (makes sense?)
This would be, after substituting, 0.1p₀ = Δhρg
We also want this pressure change to be enough for the child be able to drink the liquid (as mentioned in the question). If the length of the straw is longer than the height change (Δh) that the pressure difference can produce, the liquid will obviously not reach the top. This means the length of the straw should be equal to the height change (Δh) in order for the liquid to be drawn to the top. Hence Δh = length of straw. This is why the answer is A.
Kudos!