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A hot-air balloon carries passengers and several 1 kg sandbags, with total mass 480 kg. Its buoyancy-providing volume is V (unchanged throughout). It floats in equilibrium at height 100 m. After N sandbags are dropped, it rises to a new equilibrium at about 150 m. Air density varies as ρ(h) = ρ0*e^(-h/h0) with ρ0 = 1.25 kg/m³ and h0 = 6000 m. Find N.
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Correct answer: 4
Solution
At equilibrium the displaced-air mass equals balloon mass: m = ρ(h)V. So (480 - N)/480 = ρ(150)/ρ(100) = e^(-(150-100)/h0) = e^(-50/6000) ≈ 1 - 50/6000 = 1 - 0.00833. Thus N/480 ≈ 0.00833, N ≈ 4.
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