A spherical iron ball 10cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of `50c m^3//m in` . When the thickness of ice is 5cm, then find the rate at which the thickness of ice decreases.

Options

A. `(5)/(6pi)`
B. `(1)/(9pi)`
C. `(1)/(18pi)`
D. `(1)/(36 pi)`

Answer

C

Explanation

`(dv)/(dt) = 50 cm^(3)//"min"`
`v = (4)/(3) pi (10 + x)^(3)`
`(dv)/(dt) = (4)/(cancel(3)) pi cancel(3) (10 + x)^(2) (dx)/(dt)`
`(dx)/(dt) = (50)/(4pi (15)^(2)) = (overset(overset(cancel(2))(cancel10))cancel(50))/(underset(2)cancel(4) pi. underset(3)cancel(15).underset(3)cancel(15)) = (1)/(18pi) cm//"min"`