A cylindrical tank has base radius $5$ m and height $20$ m $.$ The tank is filled up to a height a with Liquid $x,$ of density, ${}_x=5000k\frac{g}{m^3}.$ It is then filled the rest of the way with liquid $Y,$ of density, ${}_y=2000k\frac{g}{m^3}.$ Assume that the liquids do not mix.

$($a$)$ Draw a picture of the tank, making sure to clearly label its radius, height, and heights of both liquids $x$ and $Y.$

$($b$)$ Liquid $Y$ is first moved out of the tank. Write down an expression for the work done to move liquid $Y$ out of the tank.

$($c$)$ Next, liquid $x$ is now moved out of the tank. Write down an expression for the work done to move liquid $x$ out of the tank.

$($d$)$ By equating parts $($b$)$ and $($c$),$ find a value for $a$ so that the work required to pump Liquid $x$ from the tank is equal to the amount of work required to pump liquid $Y$ from the tank.