If pressure at half the depth of a lake is equal to $\frac{2}{3}$ the pressure at the bottom of the lake, then what is the depth of the lake?

A U tube with both ends open to the atmosphere, is partially filled with water. Oil, which is immiscible with water, is poured into one side until it stands at a distance of $10 \mathrm{mm}$ above the water level on the other side. Meanwhile the water rises by $65 \mathrm{mm}$ from its original level (see diagram). The density of the oil is:

A vertical $U$-tube of uniform inner cross-section contains mercury in both sides of its arms. A glycerin (density$=1.3 \mathrm{g} {\mathrm{cm}}^{-3}$) column of length $10 \mathrm{cm}$ is introduced into one of its arms. Oil of density $0.8 \mathrm{g} {\mathrm{cm}}^{-3}$ is poured into the other arm until the upper surfaces of the oil and glycerin are in the same horizontal level. Find the length of the oil column. (Density of mercury$=13.6 \mathrm{g} {\mathrm{cm}}^{-3}$.)