Problem 7

James buys a large water bottle. Its shape (pictured below) is that of a cylinder with a conical top, ending in a small circular opening. Its total height is $35\text{cm}$, with the cylindrical portion having diameter $10\text{cm}$ and height $30\text{cm}$. The width of the opening at the top is $1\text{cm}$.

Line drawing of a cylinder, with a truncated cone placed on top, with base the top of the cylinder.

The plastic that the water bottle is made of has uniform thickness and density, and is thin enough to be modeled as a surface with mass per unit area of $0.2\text{g}\cdot\text{cm}^{-2}$. James notes that his bottle is less stable when completely full or completely empty than it is when it is partly full. Given that the density of water is $1\text{g}\cdot\text{cm}^{-3}$, how much water (in $\text{cm}^3$) should James put in the bottle to make the centre of mass as low as possible? Give your answer rounded to $4$ decimal places.

To deter guessing without thinking, we ask that you also solve the following simple arithmetic problem before checking your answer:

What is 6 × 4 + 3?

This problem was first solved on Wed 11th March at 4:16:17pm