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 (Where 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 4 meter deep to start with is -

• A. 30
• B. 45
• C. 60
• D. 80

Answer 1

The correct option is C 60

$\frac{dy}{dt}=-k\sqrt{y}{\int }_4^0\frac{dy}{\sqrt{y}}=-k{\int }_0^{t}dt2\sqrt{y}{|}_4^0=-kt=-\frac{t}{15}0-4=-\frac{t}{15}\Rightarrow t=60min$