$1)$ Water flows with velocity

v$2$v$2$ from a container $($which is open to atmosphere, has a column of water in it of height z$1$z$1$ and with cross sectional area A$1$A$1)$ through a small opening $($of area A$2$A$2)$ at the bottom. Using mass conservation, write an expression for the rate of change of the height of water v$1$v$1$ in terms of A$1,$A$2$A$1,$A$2$ and v$2$v$2.$ Note: this is a STACK question which requires an expression as an answer and all subscripts must be written with an underscore. Please do not rely on the STACK interpreter for accurate display of subscripts.

$2)$The container has a column of water in it of height z$1$z$1.$ Using Bernoulli's equation, write an expression for v$1$v$1$ in terms of v$2,$gv$2,$g and z$1$z$1.$ This is a STACK question and requires an expression on the right hand side and all subscripts must be written with an underscore. Please do not rely on the STACK interpreter for accurate display of subscripts.

$3)$The figure above shows the same container $($open to atmosphere and with a cross sectional area of A$1$A$1).$ Water is flowing out of a hole $($of area A$2$A$2)$ at the bottom with a velocity v$2$v$2.$ If the diameter at location $2$ is $10$ times smaller then the diameter of the container, which one of the following is correct?

Question $7$Select one:

a$.$

The area A$2$A$1$A$2$A$1$ so that the velocity v$2$v$2$ coming out of the container is v$2=$A$22$gz$1$v$2=$A$22$gz$1.$

b$.$

The area A$2$A$1$A$2$A$1$ so that the velocity v$2$v$2$ coming out of the container is v$2=2$gz$1$v$2=2$gz$1.$

c$.$

Due to the gravitational force per unit mass being constant we can approximate v$2=2$gz$1$v$2=2$gz$1$

d$.$

No assumptions can be made and the full equations must be solved.

$4)$What is the initial volume flow rate $($in m$33/$s to four decimal places$)$ out of the container if the initial height of water is zin$=$zin$=8$ m$,$ A$1=$A$1=21$ m$22$ and A$2=$A$2=0\mathrm{.}004$ m$22?$ If you have made any assumptions based on the geometry, these need to be included in the answer.

$5)$Assuming that A$2/$A$11$A$2/$A$11,$ write a simplified expression for the rate of change of the height of water dz$1/$dtdz$1/$dt in terms of A$1,$A$2,$gA$1,$A$2,$g and z$1$z$1.$ Note: this is a STACK question which requires an expression on the right hand side of the equals sign below and all subscripts must be written with an underscore.

dz$1$dtdz$1$dt $=$

$6)$Writing the initial water height as zinzin $($i$.$e$.$ at t$=0$t$=0$ z$1=$zinz$1=$zin$),$ give an expression for the height of the water in the cylinder as a function of time.$$If you have made any assumptions based on the geometry, these need to be included in the answer. $($Note: this is a STACK question which requires an equation as an answer and all subscripts must be written with an underscore. Please do not rely on the STACK interpreter for accurate display of subscripts. zinzin is written as z$\_$in$)$

z$1$z$1=$

$7)$How long does it take $($in seconds to two decimal places$)$ to drain a tank with A$1=$A$1=21$ m$22,$ A$2=$A$2=0\mathrm{.}004$ m$22$ and the initial height of the water $($z$1$z$1$ at t$=0)$ is zin$=$zin$=8$ m$?$