about to only mathematics

Answer

`(T=H/k)`

Explanation

Given , liquid evaporates at a rate proportional to its
surface area.
`rArr (dV)/dt propto -S …(i)`
We know that, volume of cone `=1/3pir^(2)h`
and surface area `=pir^(2)`
or ` V=1/3pir^(2)h and S=pir^(2) … (ii)`
Where, `tan theta=R/H and r/h=tan theta …(iii)`
From Eqs. (ii) and (iii), We get
`V=1/3pir^(3) cot theta and S=pi r^(2) …(iv)`
On substituting Eq. (iv) in Eq. (i), we get
`1/3 cot theta . 3r^(2)(dr)/dt=-k pir^(2)`
`rArr cot theta int_(R)^(0) dr =-k int _(0)^(T) dt`
`rArr cot theta (0-R) =-k(T-0)`
`rArr R cot theta = k T rArr H=kT [from Eq. (iii)]`
`rArr T=H/k`
`therefore Required time after which the cone is empty, T=H/k`