Some Applications Involving Separable Differential Equations

1
Some Applications Involving Separable
Differential Equations
2
Mixing Problem

• A tank contains 20 kg of salt dissolved in 5000 L
  of water. Brine that contains 0.03 kg of salt per
  liter of water enters the tank at a rate of 25 L
  / min. The solution is kept thoroughly mixed and
  drains from the tank at the same rate. Find a
  formula for the amount of salt in the tank at any
  time t. How much salt remains in the tank after ½
  hour?

3
Bacterial Growth

• A common inhabitant of the human intestines is
  the bacterium Escherichia Coli. A cell of this
  bacterium in a nutrient broth medium divides into
  two cells every 20 minutes. The initial
  population of a culture is 60 cells.
• (A) Find the relative growth rate of E. Coli.
• (B) Find an expression for the number of cells
  after t hours.
• (C) Find the number of cells after 8 hours and
  find the growth rate of the culture after 8
  hours.
• (D) When will the population of the culture reach
  20,000 cells?

4
Chemical Reactions

• In an elementary chemical reaction, single
  molecules of two reactants, X and Y, form a
  molecule of the product Z. The Law of Mass Action
  states that the rate of the reaction is
  proportional to the product of the concentrations
  of X and Y. Assuming that the initial
  concentration of X is x(0)a moles/L, the initial
  concentration of Y is y(0)b moles/L, and the
  initial concentration of Z is z(0)0 moles/L,
  find a formula for the function z(t) (which is
  the concentration of the product at any time t).
  To do this, consider two cases the case that a
  and b are different and the case that a and b are
  the same.

C
5
Radioactive Decay

• The half-life of radium-226 is 1590 years.
• A sample of radium-226 has a mass of 100 mg. Find
  a formula for the mass of radium-226 that remains
  after t years.
• Find the mass after 1000 years.
• When will the mass be reduced to 30 mg?

6
Newtons Law of Cooling

• A thermometer is taken from a room where the
  temperature is 200C to the outdoors where the
  temperature is 50C. After one minute, the
  temperature reads 120C.
• What will the reading on the thermometer be after
  one more minute?
• At what time will the thermometer read 60C?