Expert: Ahmed Salami - 3/19/2011

Question
A weather balloon with radius 9m. Springs a leak, losing air at a rate of 171 pie m^3 /min. Find the rate of decrease of the radius after 4 mins

Answer
Hi Fati,
Initial volume of balloon = 4π(9)³/3
= 972π
After 4mins, it has lost volume of air equal to 171π x 4 = 684π
Volume at 4mins = 972π - 684π = 288π = 4πr³/3  where r is the present radius
288 = 4r³/3
r³ = 216
r = 6

Volume of (spherical) balloon with radius r, V = 4πr³/3
dV/dr = 4πr²
and
dV/dt = (dV/dr) . (dr/dt)
So,
-171π = 4πr²(dr/dt)
-171 = 4(6)²(dr/dt) = 144(dr/dt)
dr/dt = -171/144
= -19/16
= -1.1875 m/min

Regards