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I found the following problem in a Math Olympiad book:

enter image description here

I know we could find the volume considering that the figure is symmetric and using a solid of revolution and an integral, the problem is that they're not giving any information about the small radius of the top of the bottle nor about the rounded part. Any suggestions to solve this problem?

Twnk
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  • Maybe pour a liquid inside? Guessing the correct shape is problematic. – Przemysław Scherwentke May 10 '18 at 18:05
  • I wonder if this uses extra information like the table here: https://en.wikipedia.org/wiki/Wine_bottle#Sizes – mvw May 10 '18 at 18:12
  • The cited question does not even put a value on r and h, so i guess the answer is not suposed to be a definite value --- it should just be something that shows you understood how to use a function as bais for a solid from revolution. just formulate the integral with f(x) with f(0)=r – bukwyrm May 11 '18 at 20:28
  • yeah but what do I do with the top radius and the curved part? I don't know if it's part of a circle, parabola, ellipse, etc... – Twnk May 11 '18 at 20:38

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enter image description here

Hints:- transform according to the question. function $1$ is $y=\frac{r}{2}$. Please scale elliptical part and bottle neck part accordingly. And the entire function is defined in $0\leq x \leq h$. Apply in the formula $V=\pi \int_{0}^{h}y^2 dx$.

  • But how did you know that the top radius is half the bottom radius and that the shape is elliptical? – Twnk May 11 '18 at 01:52
  • I haven't said anything like that. I roughly give hint for the solution. –  May 12 '18 at 16:50
  • Ok but as I stated in the question, I already knew this method. The problem is that I do not have the function that represents this figure. – Twnk May 13 '18 at 00:47