Differential Equations Exam Review (3 of 6) - Review 2...

1
Review 2: L8-L14
1. A tank holds 10,000 L and contains initially 10 kg of salt. Pure water enters the tank at
a constant rate of 5 L/min. The solution is kept well stirred and flows out at the same rate
of 5 L/min.
(a) Find the amount of salt in the tank at a time t
(b) What is the limiting amount of salt in the tank as time increases indefinitely?
2. A tank contains 10,000 L of pure water. A solution with salt concentration 0.2 kg/L
begins to enter the tank at a rate of 5 L/min. The solution is kept well stirred and flows
out at the same rate of 5 L/min.
(a) Write the initial value problem that models the rate of change of ( ) x t , where ( ) x t
the amount of salt in the tank at a time t.
(b) Find the amount of salt in the tank at a time t.
(c) What is the amount of salt in the tank in 1 hour?
(d) Find the concentration of salt
( ) C t
at a time t.
(e) When will the concentration of salt in the tank reach 0.002 kg/L?
(f) What is the limiting concentration of salt in the tank as time increases indefinitely, that
is, t → ∞
3. A tank contains 10,000 L of pure water. A solution with salt concentration 0.2 kg/L
begins to enter the tank at a rate of 5 L/min. The solution is kept well stirred and flows
out at a rate of 3 L/min.
(a) Write a differential equation that models the rate of change of
( ) x t
, where
( ) x t
is the
amount of salt in the tank at a time t.
(b) Find the amount of salt in the tank at a time t.
(c) Find the concentration of salt
( ) C t
at a time t.
(d) Assuming that the tank can hold very large amount of solution, find the limiting
concentration of salt in the tank as time increases indefinitely?
4. A certain population changes at a rate proportional to the size of the population.
Initially there were 1000 animals. In 3 years the number of animals was measured as
1200.
(a) Write a differential equation for the rate of change according to the model described
above (exponential (or Malthusian) model).
.
is
?