An open vessel is in the shape of a right-circular cone of semi-vertical angle 45 with axis

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An open vessel is in the shape of a right-circular cone of semi-vertical angle 45° with axis vertical and apex downwards. At time t = 0 the vessel is empty. Water is pumped in at a constant rate pm3s–1 and escapes through a small hole at the vertex at a rate kym3s–1, where k is a positive constant and y is the depth of water in the cone. Given that the volume of a circular cone is πr2h/3, where r is the radius of the base and h its vertical height, show that

dy ny  = p-ky dt

Deduce that the water level reaches the value y = p/(2k) at time

t =  k in 100 (In 2-3/)

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