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 $4$m deep to start with is

• A. $30$ min
• B. $45$ min
• C. $60$ min
• D. $80$ min

Answer 1

The correct option is C $60$ min

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