Gas is being pumped into a a spherical balloon at the rate of `30 ft^(3)//min`. Then the rate at which the radius increases when it reaches the value 15 ft, is

Options

A. `(1)/(30pi) ft//min`
B. `(1)/(15pi)ft//min`
C. `(1)/(20)ft//min`
D. `(1)/(25)ft//min`

Answer

A

Explanation

Let r be the radius of the spherical balloon and V be the volume at any time t.
It given that `(dV)/(dt) = 30 ft^(3)//min`
Now, `V=(4)/(3) pi r^(3)`
`implies (dV)/(dt)=4 pi r^(2)=(dr)/(dt)`
`implies ((dV)/(dt))_(r=15) = 4pi xx 15^(2)xx((dr)/(dt))_(r=15)`
`implies 30 = 900pi ((dr)/(dt))_(r=15)`
`implies ((dr)/(dt))_(r=15)=(1)/(30pi) ft//min`