Question

A farmer connects a pipe of internal diameter $20cm$ from a canal into a cylindrical tank in her field, which is $10m$ in diameter and $2m$ deep. If water flows through the pipe at the rate of $3km/h$, in how much time will the tank be filled?

Answer 1

Internal diameter of the pipe $20cm$ = $\frac{20}{100}=\frac{1}{5}m$

Internal radius$=\frac{diameter}{2}=\frac{1}{2}\times \frac{1}{5}$ =$\frac{1}{10}m$

Rate of flow of water$=3km/h$= $3000m/h$

Let the pipe take $t$ hours to fill up the tank

volume of the water that flows in $t$ hours from the pipe V(t)

$V\left(t\right)=$ Area of cross section $\times$ speed $\times$ time

$=\pi r^2\times$ speed $\times$ time

$=\pi \left(\frac{1}{100}\right)\times 3000\times t$

$V\left(t\right)=30\pi t$

Diameter of the cylinder $=10m$

radius of the cylinder $r=5m$

Depth$=2m$

Volume of the tank $V$ $=\pi r^{2}h=\pi \left(25\right)2=50\pi m^2$

Now the volume of the water that flows from the pipe in $t$ hours$=$volume of the tank

$\therefore V\left(t\right)=V$

$\therefore 30\pi t=50\pi$

$t=\frac{50}{30}$ hours

$t=\frac{50}{30}\times 60$

$t=100$ mins

$\therefore t=100$ minutes.