An open tank in the form of an inverted right circular cone has height and radius at…

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Fluids · Fluids

Problem

An open tank in the form of an inverted right circular cone has height

1.0\,\text{m}

and radius

0.80\,\text{m}

at the top. The apex of the cone is at the bottom. A small orifice of area

2.0\,\text{cm}^2

is provided at the apex. The tank is completely filled with water. (i) Show that the radius

r

of the horizontal water surface at a depth

y

measured from the apex satisfies

r = \frac{0.80}{1.0}y

. (ii) Using the result of part (i), derive a differential equation linking

\frac{dh}{dt}

and the instantaneous depth

h

of water. (iii) Hence calculate the time required to empty the tank completely. Take

g = 9.8\,\text{m s}^{-2}

and express your answer in minutes.