$(25$ pts $)$ Jenny devises a clever way to measure the unknown density of a liquid. She

builds a water tank that is divided into two parts with a partition wall as sketched. A liqui

of known density ${}_k$ is poured into the tank, and then the liquid of unknown density ${}_u$ is

carefully poured into the left half until it almost spills under the partition into the right

side. The final condition has heights as sketched. The liquids do not mix.

$($a$)$ Derive an expression for the unknown density ${}_u$ in terms of the other variables in the

problem, as needed. Stay in variables $-$ no mumbers.

$($b$)$ For ${}_k={1000}_k\frac{g}{m^3},h_0=48\mathrm{.}7cm,h_1=85\mathrm{.}3cm,$ and $h_2=99\mathrm{.}3cm,$ calculate the

unknown density.

$($c$)$ Now suppose this test is conducted on another planet with $g=15\mathrm{.}358\frac{m}{s^2},$ but the

elevations of the liquids are the same as in Part $($b$).$ What is the unknown liquid

density now? Explain.