$8$ pts$)$ Three identical objects are placed in

containers holding three different fluids with

three different densities $($r $1,$ r$2,$ and r $3$

respectively$).$ In Case $1,$ there is a string from

above connected to the object, which is

completely submerged in the fluid. In Case $2,$

there is a string connected to the bottom of the

container and the $50\%$ of the object is below

the surface of the fluid. In Case $3,$ there is no

string connected to the object, and it floats such

that $50\%$ of its volume is below the surface of the fluid. In both Case $1$ and Case $2,$ there is a non$-$

zero tension in the string.

Which of the following is the correct ranking of the densities of the three fluids?

A$.$ r$3>$ r$1>$ r$2$

B$.$ r$1>$ r$2>$ r$3$

C$.$ r$2>$ r$1>$ r$3$

D$.$ r$3>$ r$2>$ r$1$

E$.$ r$2>$ r$3>$ r