A tank, with a rectangular base and having vertical sides of height 4 m, is initially full of water. The water leaks out of a small hole in the horizontal base of the tank at a rate which, at any instant, is proportional to the square root of the depth of the water at that instant.
If x is the depth of water at time t after the leak started, write down a differential equation connecting x and t. If the tank is exactly half empty after one hour, find the further time that elapses before the tank becomes completely empty

Question

A tank, with a rectangular base and having vertical sides of height 4 m, is initially full of water. The water leaks out of a small hole in the horizontal base of the tank at a rate which, at any instant, is proportional to the square root of the depth of the water at that instant.
If x is the depth of water at time t after the leak started, write down a differential equation connecting x and t. If the tank is exactly half empty after one hour, find the further time that elapses before the tank becomes completely empty

phi · Answer

"water leaks out"  means change in volume.  Volume of the tank is the area of the base times its height.  Call the area of the base B.  Its height is 4 m.  
So 
   V= 4B
x is the depth of water at time t.   So the volume of water in the tank at time t is
   V= Bx
The change in volume wrt to time is
$$\frac{dV}{dt}= B\frac{dx}{dt}$$
"leak is  proportional to the square root of the depth"   k is some unknown constant.
$$ B\frac{dx}{dt}= k x^{1/2}$$
separate the variables:
$$ x^{-1/2} dx = \frac{k}{B} dt $$