Question

Water flows into a large tank with a flat bottom at the rate of ${10}^{-4}{m}^3{s}^{-1}$. Water is also leaking out of a hole of area $1c{m}^2$ at its bottom. If the height of the water in the tank remains steady, then this height is

• A. $5.1cm$
• B. $4cm$
• C. $2.9cm$
• D. $1.7cm$

Answer 1

The correct option is A

$5.1cm$

Step 1: Given Data

Water inflow rate $Q_{in}={10}^{-4}{m}^3{s}^{-1}$

Area of the hole $=1c{m}^2$

Let the acceleration due to gravity be $g=9.8m/s^2$.

Let the height be $h$.

Let the water outflow rate be $Q_{out}$.

Step 2: Formula Used

water inflow rate $Q_{in}=$water outflow rate $Q_{out}$

Step 3: Calculate the Height

Since the height of the water column is constant, therefore,

$Q_{out}=Q_{in}\times \sqrt{2gh}$

This is because the water flowing out is associated with a certain height acted by the acceleration due to gravity.

$\therefore Q_{out}={10}^{-4}\times \sqrt{2gh}$

Again, water inflow rate $Q_{in}=$water outflow rate $Q_{out}$

$$\begin{gathered} \therefore {10}^{-4}={10}^{-4}\times \sqrt{2gh} \\ \Rightarrow 1=2\times 9.8\times h \\ \Rightarrow h=\frac{1}{2\times 9.8}m \\ =\frac{100}{2\times 9.8}cm \\ =5.1cm \end{gathered}$$

Hence, the correct answer is option (A).