Question

Water flows along a pipe of radius 0.6 cm at 8 cm/s. How long will it take to drain out 1000$l$ of water from a tank?

Answer 1

Hint: First of all find the volume of water flowing through the pipe in one second by considering the product of the area of pipe and speed of the water. Find the area of pipe using the formula for the area of the circle, i.e. Area = $\pi {{r}^{2}}$, where r is the radius given. Now, apply the unitary method to find the time required to drain 1000$l$ of water from the tank.

Complete step-by-step solution:
Here, we have been provided with a pipe whose radius is 0.6 cm. Water is flowing through it at a speed of 8 cm per second. We have to find the time it will take to drain 1000$l$ of water from a tank.
Now, the pipe must have a circular part at its end. So, its area will be given as Area = $\pi {{r}^{2}}$, where r is the given radius. So, we have
$\begin{align}
  & \Rightarrow Area=\pi {{\left( 0.6 \right)}^{2}} \\
 & \Rightarrow Area=0.36\pi c{{m}^{2}} \\
\end{align}$
Now, the volume of water flowing through the pipe will be the product of the area of the circular part and the speed of the water. So, we have
$\Rightarrow $ Volume of water flowing through the pipe per second
$\begin{align}
  & \Rightarrow 0.36\pi \times 8c{{m}^{3}} \\
 & \Rightarrow 2.88\pi c{{m}^{3}} \\
\end{align}$
So, the pipe will take 1 second to drain out $2.88\pi c{{m}^{3}}$ of volume. Therefore, we can say that using the unitary method,
Time taken to drain $2.88\pi c{{m}^{3}}$ volume = 1 second
Time taken to drain $1c{{m}^{3}}$ volume = $\dfrac{1}{2.88\pi }$ seconds
Applying the conversion, $1c{{m}^{3}}=1ml=\dfrac{1}{1000}l$ , we get
Time taken to drain $\dfrac{1}{1000}l$ volume = $\dfrac{1}{2.88\pi }$ seconds
Time taken to drain 1 l volume = $\dfrac{1000}{2.88\pi }$ seconds
Time taken to drain 1000 l volume = $\dfrac{1000\times 1000}{2.88\pi }$ seconds
$\Rightarrow \dfrac{1000\times 1000}{2.88\pi \times 3600}$ hours
$\Rightarrow \dfrac{1000\times 10}{2.88\times 3.14\times 36}$ hours
$\Rightarrow 30.72$ hours
Therefore, the given pipe will take 30.72 hours to drain the provided tank.

Note: One may note that we have substituted $\pi =3.14$ in the last step of calculation of time. There is no information provided regarding the value of $\pi $, so one can also use $\pi =\dfrac{22}{7}$ as it will not alter the answer much. One may see that we have changed the time from seconds to hours by using the relation - $1s=\dfrac{1}{3600}hr$. It was necessary because we were getting a large number in terms of seconds. One must remember the conversion formula $1c{{m}^{3}}=1ml$ to solve the above question.