Which fundamental law forms the basis of the equation of continuity?

Water falls from a tap,down the streamline.

The figure below shows a liquid being pushed out of a tube by a piston having a cross-sectional area of $2.0\,cm^2$. The cross-sectional area at the outlet is $10\,mm^2$. If the piston is pushed at a speed of $4\,cm/s$,the speed of the outgoing fluid is $.........\,cm/s$.

$A$ river gradually deepens from a depth of $4 \ m$ to a depth of $8 \ m$ as shown. The width,$W$,of the river does not change. At the depth of $4 \ m$,the river's speed is $12 \ m/s$. Its velocity at the $8 \ m$ depth is ......... $m/s$.

The surface of water in a water tank of cross-sectional area $750 \,cm^2$ on the top of a house is '$h$' $m$ above the tap level. The speed of water coming out through the tap of cross-sectional area $500 \,mm^2$ is $30 \,cm/s$. At that instant, $\frac{dh}{dt}$ is $x \times 10^{-3} \,m/s$. The value of '$x$' will be:

Water is moving with a speed of $5.18 \; ms^{-1}$ through a pipe with a cross-sectional area of $4.20 \; cm^2$. The water gradually descends $9.66 \; m$ as the pipe increases in area to $7.60 \; cm^2$. The speed of flow at the lower level is ....... $ms^{-1}$.