## Related questions with answers

In Exercise, use the differential equation for a leaking container, Eq. $\frac{d y}{d t}=-\frac{B \sqrt{2 g y}}{A(y)}$

At t=0, a conical tank of height 300 cm and top radius 100 cm [Figure] is filled with water. Water leaks through a hole in the bottom of area $B=3 \mathrm{~cm}^2$. Let y(t) be the water level at time t.

(a) Show that the tank's cross-sectional area at height y is $A(y)=\frac{\pi}{9} y^2$.

(b) Solve the differential equation satisfied by y(t)

(c) How long does it take for the tank to empty?

Solution

VerifiedThe model for a leak of a water from the container that has a hole of area $B$ in the bottom, and $A(y)$ is the area of a horizontal cross section at the water level $y$ is

$\frac {dy } {dt }=-\frac {B\sqrt { 2gy} } {A(y) }$

where $g$ is acceleration due to gravity.

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