If a cylinder tank holds 100,000 liters of water, which can be drained from the bottom of the tank in an hour, then Toricellis law gives the volume V of water r

Question

If a cylinder tank holds 100,000 liters of water, which can be drained from the bottom of the tank in an hour, then Toricellis law gives the volume V of water remaining in the tank after T minutes as

 

V(T) = 100,000(1 - (1/60)t)^2   0 ≤ t ≤ 60

 

Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to T) as a function T. What are its units? For times T = 0, 10, 20, 30, 40, 50 and 60 minutes. Find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two. At what time is the flow rate greatest? The least?

 

Answer 1

Hi Abegail:

This is how to tackle this type of problem.

We are given that: V(t) = 100,000(1 - t/60)^2  for 0 ≤ t ≤ 60

The rate of change of V is just the derivative, dV/dt

By the Chain Rule, dV/dt = 100,000 . 2 (1 - t/60) . (-1/60)

= -3,333.33 (1 - t/60)

The units are liters per minute. Since this is an outflow, dV/dt is negative.

At t=0, dV/dt = -3,333.33 L/min [plugging t=0 into the above equation

At t=10, dV/dt = -3,333.33(1 - 10/60) = -3,333.33 (5/6) = 2,777.78 L/min

At t=20, dV/dt = -3,333.33 (1 - 20/60) =-3,333.33 (2/3) = 2,222. 22 L/min

And so on for the other times.