In lecture $2$ we solved the barometric equation for pressure as a function of height $z$ :

$\frac{dP}{dh}=-g$

where $p$ is pressure, $$ is density, and $g$ is the acceleration due to gravity.

a$.$ Suppose that density is linearly proportional to pressure. Follow the derivation in class and solve the differential equation to show that $p(h)=p_0{e}^{-\frac{h}{h_0}},$ where the scale height is $h_0=\frac{p_0}{{}_0}g.$

b$.$ If the height $h$ is much smaller than the scale height $h_0,$ show that $p(h)$~~ $p_0(1-\frac{h}{h_0}).$ Explain the physical significance of this result.

$4.$ A treasure chest with a volume $V=0\mathrm{.}0500m^3$ lies at the bottom of a lake whose water has a density of $1\mathrm{.}00{10}^{3}k\frac{g}{m^3}.$ How much force is required to lift the chest, if the mass of the chest is $1000kg?$