Question

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

Answer 1

Step 1: Find dimensions of the pipe and the tank.

Radius $\left(r_1\right)$ of circular end of pipe $=\frac{20}{2\times 100}=0.1m$
Radius $\left(r_2\right)$ of circular end of cylindrical tank $=\frac{10}{2}=5m$

Depth $\left(h_2\right)$ of cylindrical tank $=2m$

Step 2: Find Volume of water flowing in t minutes from pipe.

Area of cross-section of pipe
$=\pi \times r_1^2=\pi \times (0.1{)}^2=0.01\pi m^2$
Speed of water $=3\frac{km}{hour}$
$=\frac{3000}{60}=50\frac{m}{min}$.
Volume of water flowing in $1$ min from pipe $=$ Distance travelled by water in $1$ min $\times$ Area of cross section of the pipe $=50\times 0.01\pi$
$=0.5\pi m^3$
Volume of water flowing in $t$ mins from pipe $=t\times 0.5\pi m^3$

Step 3: Find time taken to fill the tank.

Let the tank be filled completely in $t$ minutes.
$\therefore$ Volume of water flowing in $t$ minutes from pipe
$\Rightarrow$ Volume of water in tank
$\Rightarrow t\times 0.5\pi =\pi r_2^2\times h_2$
$\Rightarrow t\times 0.5=5^2\times 2$
$\Rightarrow t=100mins$

So the cylindrical tank will fill in $100$ minutes.