Question

Water flows at the rate of $10$ metre per minute through a cylindrical pipe having its diameter $30$ $mm$. How much time will it take to fill a conical vessel of base diameter $40$ cm and depth $24$ $cm$?

Answer 1

Water flows through a cylindrical pipe of diameter $30$ mm at the rate of $10$ m per minute.

$\therefore$ Volume of water flowing out in $1$ minute $=$ Volume of cylinder of length $10$ m and diameter $30$ mm.

Here radius, $r=\frac{30}{2}=15$ mm $=1.5$ cm
and length, $h=10$ m $=1000$ cm.

$\therefore$ Volume of water flowing out in $1$ minute
$=\pi r^{2}h$

$=\pi \times {\left(1.5\right)}^2\times 1000$

$=2250\pi {cm}^3$

Now, diameter of conical vessel $=40$ cm.

$\therefore$ Its radius, $r_1=20$ cm

and Depth $=$ height, $h_1=24$ cm

$\therefore$ Volume of vessel $=\frac{1}{3}\pi r_1^2{h}_1$

$=\frac{\pi }{3}\times {\left(20\right)}^2\times 24$

Suppose the conical vessel is filled in $t$ minutes.

$\therefore$ Volume of water flowing out in $t$ minutes $=$ Volume of conical vessel

$\therefore$ $2250\pi \times t=\frac{\pi }{3}\times {\left(20\right)}^2\times 24$

$\therefore t=\frac{1}{3}\times \frac{20\times 20\times 24}{2250}$ .....[$\because$ $\pi$ is cancelled out]

$=1.42$ minute.

Thus, $1.42$ minute will be required to fill the conical vessel.