3650.41 – Pouring Between Glasses

When the liquid in a certain cylindrical glass is poured into a second cylindrical glass whose radius is 2 inches greater, the height reached in the second glass is one half of that reached in the first. Find, to the nearest tenth of an inch, the radii of the two glasses.

Solution

Referring to the figure below, we find that,

\begin{aligned} \pi r^2 (2h) &=& \pi (r+2)^2 h, \\ &=& \pi (r^2 + 4r +4) h, \\ 2r^2 &=& r^2 +4r+4, \\ r^2 - 4r -4 &=& 0. \end{aligned}

Solving for $r$ ,

$r = \frac{4 ± \sqrt{16 - 4(-4)}}{2} = 2 ± 2\sqrt{2}.$

Let's go with $r = 2 + 2\sqrt{2} \approx 4.8 \text{ in,}$ as the smaller radius, making the larger one approximately $6.8$ in.