Flipping a Cone

The formula for the height of a right circular cone is $\frac{h \pi r^2}{3},$ where $h$ is the height of the cone and $r$ is the radius of the cone. Therefore, the total volume of the cone is $\frac{100 \pi r^2 }{3}$ and the volume of the water is $\frac{25 \pi \left(\frac{r}{4}\right)^2}{3} = \frac{25 \pi r^2}{3 \times 16}.$ Therefore, the volume of the empty space in the cone is $\frac{1600 \pi r^2 }{3 \times 16} - \frac{25 \pi r^2}{3 \times 16} = \frac{1575 \pi r^2 }{3 \times 16} = \frac{525 \pi r^2 }{16}.$

When the cone is flipped, this volume of empty space moves to the apex-end of the cone container, but it still has the same volume. Therefore, we can solve backwards for the new height of the empty space. Note that if the height is $h,$ the radius of the cone at that height will be $\frac{rh}{100}$ where \(r) is the unknown radius of the container.

\[\frac{525 \pi r^2 }{16} = \frac{h \pi (\frac{rh}{100})^2}{3}\] $\frac{525 r^2 }{16} = \frac{h^3 r^2}{30000}$ $\frac{525}{16} = \frac{h^3}{30000}$ $984375 = h^3$ $99.476 ~ h$

Therefore, the height of the water is $100 - 99.476 = .524,$ so about .524 centimeters.