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)cm//min`
B. `(1)/(54pi) cm//min`
C. `(1)/(18pi) cm//min`
D. `(1)/(36pi) cm//min`

Answer

C

Explanation

Let at any time, h cm be the thickness of ice. Then, V=Volume of ice `=(4)/(3)pi(10+h)^(3)-(4)/(3)pixx10^(3)`
`implies (dV)/(dt)=4pi(10+h)^(2)(dh)/(dt)`
`implies -50=4pi (10+5)^(2)xx(dh)/(dt) " "[because (dv)/(dt)=-50 cm^(3)//min and h = 5]`
`implies (dh)/(dt) = -(1)/(18pi) cm//min`