Question

Figure shows a large container containing water to height 'H = 8 m'. The front portion of the container has a rectangular portion cut out from it. Because of that water comes out of it causing the water level to reduce at a certain rate. How many times (let's call it 'n') will this rate decrease when its level changes from H = 8 m to h = 2 m. Fill η/4 in OMR sheet.

Answer 1

1/2

Answer 2

The rate of water flow out of an opening at depth $h$ is proportional to $\sqrt{h}$ (Torricelli's law). The rate of decrease of the water level, $-\frac{dh}{dt}$, is proportional to the rate of outflow divided by the cross-sectional area of the container. Assuming the cross-sectional area of the container and the area of the opening are constant with respect to height, the rate of decrease of water level is proportional to $\sqrt{h}$.

Let the rate of decrease be $R(h) = C\sqrt{h}$, where $C$ is a constant.

The rate at $H=8$ m is $R_1 = C\sqrt{8}$. The rate at $h=2$ m is $R_2 = C\sqrt{2}$.

The factor 'n' by which the rate decreases is $n = \frac{R_1}{R_2} = \frac{C\sqrt{8}}{C\sqrt{2}} = \sqrt{\frac{8}{2}} = \sqrt{4} = 2$.

The question asks to fill $\eta/4$ in the OMR sheet, where $\eta$ is the factor 'n'. So, $\eta = 2$. The value to be filled is $\eta/4 = 2/4 = 1/2$.