A spherical balloon is filled with 4500p cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of `72pi` cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is (1) `9/7` (2) `7/9` (3) `2/9` (4) `9/2`

Options

A. `6//7`
B. `4//9`
C. `2//9`
D. None of these

Answer

C

Explanation

`V=4/3pir^(3)`
Initially volume of gas in balloon is 4500 `pi` cubic meters.
`therefore 4500 pi=(4pir^(3))/(3)`
`therefore r=15` m.
Given `(dV)/(dt) = 72pi` cubic meter per minute
`therefore` After 49 minutes, volume of gas leaked
`=72pi xx 49`
`therefore` Volume of gas in balloon after 49 minutes
`=(4500-49xx72)pi=972pim^(3)`
`therefore 972pi=(4pir^(3))/(3)`
`therefore r^(3)=3 xx 243 = 3 xx 3^(5)`
`therefore r=9`
Now, `(dV)/(dt) = 4pir^(2) (dr)/(dt)`
For the rate of decrease in radius after 49 minutes.
`72pi = 4pi xx 9 xx 9(dr)/(dt)`
`(dr)/(dt) 2/9` meter per minute