## 9. Water, Oil, and Mercury

In a conical measuring cylinder, mercury (density 13.59), water (density 1), and oil (density 0.915) are poured sequentially. The three liquids fill the glass without mixing and form three layers of equal thickness. Which of the liquids poured into the measuring cylinder has the greatest mass?
9. The volume of a cone is equal to one third of the product of its height and the area of its base. But given the angle at the vertex of the cone, the radius of the circle lying at the base is proportional to the height of the cone. Therefore, the volume of the cone is proportional to the cube of its height.

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-154.jpg?height=614&width=282&top_left_y=561&top_left_x=887)

Let \( V_{\mathbf{1}} \) be the volume of mercury, \( V_{2} \) be the volume of water, and \( V_{3} \) be the volume of oil; \( l \) be the height of each layer, and \( a \) be the proportionality coefficient of the volume of the cone to the cube of its height.

Consider sequentially three cones of heights \( l \), \( 2l \), and \( 3l \).

\[
\begin{gathered}
V_{\mathbf{1}} = a l^{3} \\
V_{\mathrm{i}} + V_{\mathbf{2}} = a(2 l)^{3} = 8 V_{\mathrm{i}} \\
V_{\mathbf{1}} + V_{\mathbf{2}} + V_{\mathbf{3}} = a(3 l)^{3} = 27 V_{\mathrm{i}}
\end{gathered}
\]

From this, \( V_{2} = 7 V_{1} \) and \( V_{3} = 19 V_{1} \). Then

mass of mercury \( = 13.59 V_{1} \),

mass of water \( = 1 \cdot V_{2} = 7 V_{1} \),

mass of oil \( = 0.915 V_{3} = 17.385 V_{1} \).

Therefore, there is the most oil in the measuring cylinder.