Question

The inner length and the
breadth of a rectangular sump
are 20 m and 15 m respectively.
Water flows through an inlet
pipe of cross-sectional area 0.25
sqm, at the rate of 200 m per
minute. If the total time taken to
fill the tank is one hour, find the
depth of the sump​

Answer 1

First we find the volume rate, i.e., volume of water flown through the pipe per unit time.

The volume rate is actually equal to the cross sectional area of the pipe and the speed of water flow, i.e.,

$\longrightarrow\sf{\dfrac{V}{t}=0.25\times200\ m^3/min}$

$\longrightarrow\sf{\dfrac{V}{t}=50\ m^3\,min^{-1}\quad\quad\dots(1)}$

But,

$\longrightarrow\sf{1\ hr=60\ min}$

$\longrightarrow\sf{1\ min=\dfrac{1}{60}\ hr}$

$\longrightarrow\sf{1\ min^{-1}=\left(\dfrac{1}{60}\right)^{-1}\ hr^{-1}}$

$\longrightarrow\sf{1\ min^{-1}=60\ hr^{-1}}$

Then (1) becomes,

$\longrightarrow\sf{\dfrac{V}{t}=50\times60\ m^3\,hr^{-1}}$

$\longrightarrow\sf{\dfrac{V}{t}=3000\ m^3\,hr^{-1}}$

$\longrightarrow\sf{V=3000t\ m^3}$

Given that the sump gets fully filled for one hour. So,

• $\sf{t=1\ hr}$

Then the volume of the sump is,

$\longrightarrow\sf{V=3000\times1\ m^3}$

$\longrightarrow\sf{V=3000\ m^3}$

Given that the length and breadth of the sump are $\sf{20\ m}$ and $\sf{15\ m}$ respectively.

Let the depth of the sump be $\sf{h.}$ Hence we have, volume of the sump,

$\longrightarrow\sf{V=lbh}$

$\longrightarrow\sf{3000=20\times15h}$

$\longrightarrow\sf{3000=300h}$

$\longrightarrow\sf{h=\dfrac{3000}{300}}$

$\longrightarrow\sf{\underline{\underline{h=10\ m}}}$

Hence the depth of the sump is $\bf{10\ m.}$