When a balloon rises, the air pressure drops to half the air pressure at sea level p$\_0=1\mathrm{.}01325·10^5$ Pa$.$ The density of the air on the ground is $$rho $\_0=1,293$kg$/$m$^3.$ The temperature of the atmosphere and the balloon should be assumed to be constant, so that the product p $·$ V can be assumed to be constant. 

$($a$)$ At what altitude is the balloon? $$

$($b$)$ What density $$rho does the air have at this altitude?

$($c$)$ What maximum climbing height will the balloon reach if it is filled with gas of density $$rho $\_$G$(0)$ to the volume V$\_0$ at sea level and then tightly closed? The mass of the solid balloon envelope is m$\_$B and that of the payload is m$\_$N$.$ The volume of the balloon envelope and payload can be neglected.