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January 26th, 2010, 07:41 PM  #1 
Senior Member Joined: Jan 2009 Posts: 345 Thanks: 3  Write a differential equation for V at time t
According to Torricelli's law the time rate of change of the volume V of a water draining tank is proportional to the square root of the water's depth. A cylindrical tank of radius centimeters and height 16 centimeters, which was full initially, took 40 seconds to drain. (a) Write a differential equation for V at time t and the two corresponding conditions. (b) Solve the differential equation. (c) Find the volume of water after 10 seconds. 
January 27th, 2010, 02:25 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 21,130 Thanks: 2339 
If the water depth at time t is h, dV/dt = k?h, where k is a negative constant and 0 seconds ? t ? 40 seconds. V = (100?²cm²)h. When t = 0 seconds, h = 16cm, so V = 1600?²cm³; when t = 40 seconds, h = 0cm, so V = 0cm³. ?V= (10?cm)?h, so (10?cm)(dV/dt)/?V = k. Have a go at integrating that (it's convenient to use integration limits of 40 seconds and t) and finishing the question, posting your work so that we can check it. When evaluating k, take care to give it the right units. 

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