A spherical balloon is filled with 4500 cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72 cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage has begun, is

Step-by-step solution

Volume analysis for spherical balloon with escaping helium : The volume V and radius r obey V = 4 3 r^3 . Initial volume V₀ = 4500 m³ decreases at a constant rate dV dt = -72 m³/min due to leakage. After 49 minutes, the remaining volume is V₄₉ = V₀ - 72 49 = 4500 - 3528 = 972 m³. This corresponds to radius r satisfying 972 = 4 3 r^3 , or r^3 = 972 3 4 = 729 , hence r = 9 m. Differentiating the volume relation implicitly yields dV dt = 4 r^2 dr dt . Substituting dV dt = -72 and r = 9 gives -72 = 4 (81) dr dt . Solvi