A hot air 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 buoyancy 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 ρ$$(h)=$$ρ$$0e$$−$$hh0$$, where ρ$$0=1.25kgm$$−$$3$$ and $$h0=6000$$ m, the value of N is $$_________$$.

Question

A hot air 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 buoyancy 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 ρ$$(h)=$$ρ$$0e$$−$$hh0$$, where ρ$$0=1.25kgm$$−$$3$$ and $$h0=6000$$ m, the value of N is $$_________$$.

Infinity Learn · Accepted Answer

$$480$$×$$g=V$$ρ$$1g(480$$−$$N)g=V$$ρ$$2g$$∴$$480$$−$$N480=$$ρ$$2$$ρ$$1$$⇒$$1$$−$$N480=e$$−$$h2/h0e$$−$$h1/h0=eh1$$−$$h2h0=e-506000$$⇒$$1$$−$$N480=1$$−$$506000$$∵If x>$$0$$ then $$ex=1+x+x22$$!$$+---$$ and If x<$$0$$ then $$ex=1-x+x22$$!$$+---$$ ⇒$$N=50$$×$$4806000=4$$

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