A helium balloon has its rubber envelope of weight `w_("Rubber")` and its is used to lift a weight of `w_("Load")`. The volume of fully inflated balloon is `V_(0)`. The atmospheric temperature is `T_(0)` throughout and the atmospheric pressure is known to change with height y according to equation `P = P_(0) e^( – ky)` where k is a positive constant and `P_(0)` is atmospheric pres- sure at ground level. The balloon is inflated with sufficient amount of helium so that the net upward force on is `F_(0)` at the ground level. Assume that pressure inside the balloon is always equal to the outside atmospheric pressure. Density of air and helium at ground level is `rho_(a)` and `rho_(He)` respectively. (a) Find the number of moles (n) of the helium in the balloon. For following two questions assume n to be a known quantity. (b) Find the height `(y(0))` at which the balloon is fully inflated. (c) Prove that the balloon will be able to rise to height `y_(0)` if `(nRT_(0)g)/(P_(0))(rho_(a)-rho_(Hg)) gt w_("Rubber"+w_("Load")`