Question

Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth $y$, where the constant of proportionality $k>0$ depends on the acceleration due to gravity and the geometry of the hole. If $t$ is measured in minutes and $k=\frac{1}{15}$, then the time to drain the tank if the water is $4\text{m}$ deep to start with is

• A. $30\text{min}$
• B. $45\text{min}$
• C. $60\text{min}$
• D. $80\text{min}$

Answer 1

The correct option is C $60\text{min}$

According to the equation,
$\frac{dy}{dt}=-k\sqrt{y}$
or $\underset{4}{\overset{0}{\int }}\frac{dy}{\sqrt{y}}-k\underset{0}{\overset{t}{\int }}dt=0$
or $2\sqrt{y}{|}_4^0=-kt=-\frac{t}{15}$
or $0-4=-\frac{t}{15}$
or $t=60\text{min}$