A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant $= k > 0$ ). Suppose that $r (t)$ is the radius of liquid cone at time t.
The time after which the cone is empty is

Question

A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant $= k > 0$ ). Suppose that $r (t)$ is the radius of liquid cone at time t.

The time after which the cone is empty is

TestHub · Accepted Answer

Let θ be the semi vertical angle of the cone so that tan θ = R / H . Let the radius and height of water cone at time t be r and h respectively. So tan θ = h r ​ ​ V = 3 1 ​ π r 2 h = 3 1 ​ π r 3 cot θ and S = π r 2 ​ d t d V ​ = − k S ⇒ 3 1 ​ π ( 3 r 2 ) cot θ d t d r ​ = − k ( π r 2 ) ⇒ d t d r ​ = − k tan θ ​ Integrating, r = − ( k tan θ ) t + C . When t = 0 , r = R , ∴ C = R . Thus r = − ( k tan θ ) t + R When cone is empty, r = 0 ⇒ T = k t a n θ R ​ = k ( R / H ) R ​ = k H ​

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