A hot balloon is carrying some passengers, and a few sandbags of mass 1 kg each so that its total mass is 480 kg. Its effective volume giving the balloon its bouncy is V. The balloon is floating at an equilibrium height of 100 m. When N number of sandbags are thrown out, the balloon rises to a new equilibrium height close to 150 m with its volume V remaining unchanged. If the variation of the density of air with height h from the ground is $\rho(h) =\rho_0\,e^{{h}/{h_0}}$, where $\rho_0=1.25\,kgm^{-3}$ and $h_0=6000\,m$ the value of N is ________.

Hint: In this question we use a formula for the buoyant force which is applied in an upward direction. Also, the weight of any object is always in a downward direction. Hence both forces must be equal to attain the equilibrium condition.

Formula used:
Archimedes’ principle is given as,
\[{F_b} = \rho Vg\]
Where \[{F_b}\]is the buoyant force, \[\rho \]is the fluid density, g is the acceleration due to gravity and V is fluid volume.

Complete step by step solution:
As we know Archimedes’ principle is ,
\[{F_b} = \rho Vg\]
\[\Rightarrow mg = \rho Vg\]
Initial condition when height is 100 m, putting the values
\[480 \times 10 = {\rho _0}{e^{\dfrac{h}{{{h_0}}}}}Vg\]
\[\Rightarrow 480 \times 10 = {\rho _0}{e^{ - \dfrac{{100}}{{6000}}}}Vg\] …..1
The final condition is when the height is 150 m, putting the values
\[(480 - N) \times 10 = {\rho _0}{e^{ - \dfrac{{150}}{{6000}}}}Vg\] ……2
Divide equations 1 and 2,
\[\dfrac{{480}}{{480 - N}} = {e^{ - \dfrac{{100}}{{6000}} - \left( {\dfrac{{150}}{{6000}}} \right)}}\]
By solving this, we get
$\therefore N=4$

Therefore, the value of N is 4.

Note: Buoyant force is not only creating an upward lift on an object which is in a fluid. It is also equal to the weight of the fluid displaced by that object. This was discovered by Archimedes. So, this is also known as Archimedes’ Principle. It is very important to know that it is about fluids.