Water Rising in a Tank

Date: Wed, 3 Jul 1996 18:22:11 -0400 (EDT) 
From: Anonymous
Subject: How fast is the water level rising?

A water tank is 30 meters wide, 60 meters long, and is 10 meters 
deep at one end with the depth increasing linearly until it is 15 
meters deep at the other end. Water is run into the tank at the rate 
of 5 cubic meters per minute. How fast is the water level rising 
when it is 3 meters deep?

Thanking you ahead of time, Dr. Math.

Date: Thu, 4 Jul 1996 12:43:49 -0400 (EDT) 
From: Dr. Anthony
Subject: How fast is the water level rising? 

I take this to mean 3 metres at the deep end so we still have the 
sloping floor exposed.

The sloping part of the tank has a flat surface 'length' 12 times the 
depth, so if the depth is x and the length is 12x, the width is a 
constant 30 metres. The volume of water present when the depth is 
x is therefore: 

V = (1/2)(x)(12x)*30 = 180*x^2 so dV/dx = 360x 

Now water is entering at 5 m^3 per min, so dV/dt = 5, and we 
require dx/dt. 

We have dx/dt = dx/dV * dV/dt

= 1/(360x) * 5

= 1/(72x)

Now when x = 3, this gives dx/dt = 1/216 metres per minute 


-Doctor Anthony, The Math Forum