Question

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

Answer 1

Hint: The volume of water which flows into the tank is the same as flown out of the pipe. As both the pipe and the tank are cylindrical in structure, the volume of them can be calculated from the standard volume formula.

Complete step-by-step answer:

We know that the water flows from the pipe into the tank. Let the time taken to fill the tank be x hours.
As the cross section of the pipe would be a circle with diameter 20cm=0.2m, its radius will be r=0.1m. The cross-sectional area of the pipe would be equal to $\pi {{r}^{2}}=\pi \times {{\left( 0.1 \right)}^{2}}=0.01\pi \text{ }{{\text{m}}^{2}}$
The distance travelled by the water in x hours is equal to speed of water multiplied by the time taken. Thus,
$\text{Distance travelled by the water in x hours = 3km/hr }\times \text{ x hours=3x km=3000x m}...............\text{(1}\text{.1)}$
Now, the volume of the water flown into the tank would be the cross sectional area times the length covered by the water in x hours. Thus, the volume of the water flown into the tank in x hours should be:
$\begin{align}
  & \text{Volume of water flown into the tank=distance travelled by water}\times \text{cross sectional area} \\
 & \text{=3000x m }\times 0.01\pi {{\text{m}}^{2}}=30\pi x\text{ }{{\text{m}}^{3}} \\
\end{align}$
As the tank is in the form of a cylinder with radius of base= $\dfrac{diameter}{2}=\dfrac{20m}{2}=10m$ and height 2m, its volume would be given by
 $\text{Volume of the tank=}\pi {{\text{r}}^{2}}h=\pi {{\left( 10 \right)}^{2}}\times 2=200\pi \text{ }{{\text{m}}^{3}}$
As when the tank is filled, the volume of the water flown into the tank would be equal to the volume of the tank, we should have
$\begin{align}
  & \text{Volume of water flown in x hours=Volume of tank} \\
 & \Rightarrow \text{30}\pi \text{x=200}\pi \Rightarrow x=\dfrac{200}{30}hours=\dfrac{200}{30}\times 60\text{minutes=400minutes} \\
\end{align}$
Thus, the time taken to fill the tank is 400minutes or $\dfrac{20}{3}hours$.

Note: We should be careful to convert the time into meters in equation (1.1) as all the length units should be converted into the same units while solving the problem.