A large balloon is designed to float without any motors at a height of $200$ m $,$ carrying a payload of total mass $\backslash (3,000\backslash$mathrm$\{$~kg$\}\backslash ).$ The balloon contains helium gas, whose specific volume at $\backslash (15^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ is $\backslash (6\mathrm{.}0\backslash$mathrm$\{$~m$\}^\{3\}\backslash )$ per kg $.$ The specific volume of air at $\backslash (15^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ is $\backslash (0\mathrm{.}82\backslash$mathrm$\{$~m$\}^\{3\}\backslash )$ per kg $.$

a$)$ Explain the principle behind the force that allows the balloon to float and support the weight of its payload.

b$)$ Calculate the volume that the balloon will have when floating at a height of $200$ m $.$

c$)$ How much helium gas does the balloon need to contain to float at this height?

d$)$ Explain how your answers above would be different if the temperature of the air were lower, or if the air pressure were higher.