$(13)$ A square glass tank with side length $10$ m is half filled with water.

$(15$ marks$)$

A solid metal cylinder of radius $2$ m and height $12$ m is being pushed vertically into the centre of the tank water as in the picture below.

At any time, the speed at which the cylinder is being lowered into the the tank $($measured in meters per minute$)$ is the same as the distance between the base of the cylinder and the bottom of the tank $($measured in meters$).$

Water Level: The water level is the height of the water from the bottom of the cube.

Hint: The volume $\backslash ($ V $\backslash )$ of a cylinder with any base is given by $\backslash ($ V$=$A$\_\{$b$\}\backslash$cdot h $\backslash ),$ where $\backslash ($ A$\_\{$b$\}\backslash )$ is the area of the base and $\backslash ($ h $\backslash )$ the height of the cylinder.

$(6$ marks$)$ How quickly is the water level on the side of the tank rising when the base of the cylinder is $4$ m from the bottom of the tank?