Question

Water flows at the rate of $10\mathrm{m}/\mathrm{minute}$ through a cylindrical pipe $5\mathrm{mm}$ in diameter. How long would it take to fill a conical vessel whose diameter at the base is $40\mathrm{cm}$ and depth $24\mathrm{cm}$?

Answer 1

Step 1: Writing the given information

Base Diameter of conical pipe$=40\mathrm{cm}$

Height of conical pipe, $\mathrm{h}=24\mathrm{cm}$

Base diameter of cylindrical pipe$=5\mathrm{mm}=0.5\mathrm{cm}[\mathrm{Since}1\mathrm{cm}=10\mathrm{mm}]$

Rate of flow of water is height of cylinder i.e. $10\mathrm{m}/\mathrm{minute}=1000\mathrm{cm}/\mathrm{minute}$.

Step 2: Finding the volume of conical vessel

$\begin{array}{rcl}\mathrm{Volume}\mathrm{of}\mathrm{cone}& =& \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}\\ & =& \frac{1}{3}\mathrm{\pi }{\left(20\right)}^2\left(24\right)\\ & =& 3200\mathrm{\pi }{\mathrm{cm}}^3\end{array}$

In cylindrical pipe,

Let $\mathrm{t}$ be the time taken to fill the conical vessel.

Water covers $10\mathrm{t}\mathrm{m}=1000\mathrm{t}\mathrm{cm}$ distance in pipe,

Step 3: Finding the volume of cylindrical pipe

$\begin{array}{rcl}\mathrm{Volume}\mathrm{of}\mathrm{water}\mathrm{passed}\mathrm{in}\mathrm{pipe}& =& \mathrm{Volume}\mathrm{of}\mathrm{a}\mathrm{cylinder}\\ & =& {\mathrm{\pi r}}^2\mathrm{h}\\ & =& \mathrm{\pi }{(0.25)}^2\left(1000\mathrm{t}\right)\\ & =& 62.5\mathrm{t\pi }{\mathrm{cm}}^3\end{array}$

Step 4: Finding the time taken to fill a conical vessel from cylindrical pipe

$$\begin{gathered} \mathrm{Volume}\mathrm{of}\mathrm{conical}\mathrm{vessel}=\mathrm{Volume}\mathrm{of}\mathrm{water}\mathrm{that}\mathrm{passes}\mathrm{through}\mathrm{pipe}\mathrm{in}\mathrm{t}\mathrm{minutes} \\ \Rightarrow 62.5\mathrm{t\pi }=3200\mathrm{\pi } \\ \Rightarrow 62.5\mathrm{t}=3200 \\ \Rightarrow \mathrm{t}=51.2\mathrm{minutes} \end{gathered}$$

We know that,

$0.2\mathrm{minutes}=0.2\left(60\right)\mathrm{seconds}=12\mathrm{seconds}[\mathrm{Since}1\mathrm{minute}=60\mathrm{seconds}]$

Therefore, $\mathrm{t}=51\mathrm{minutes}12\mathrm{seconds}$

Therefore, the time taken to fill a conical vessel from cylindrical pipe, $\mathrm{t}=51\mathrm{minutes}12\mathrm{seconds}$