Exponential growth and decay
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Exponential growth and decay

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Exponential growth and decay Exponential growth and decay Presentation Transcript

  • Laws of Growth and Decay
  • Natural Growth and Decay dP = kP → P = P0 e ! kt dt
  • Example 1• The annual growth rate of a town is 3% of its population. In 2000 the population is 13000. • Show that the population can be calculated by the general growth and decay model • Hence find the population in 2010 (to nearest 100)
  • Example 2 - Half LifeThe half life a a radio-active substance is 200years. Initially there are 200g. How muchmass remains after 300 years.
  • Law of Limited Growth and DecayIn reality the population growth may berestricted due to death, diseases orlimitations in food, water etc DP If = k(P − A) → P = A + Be ! kt dt
  • Example 3 - NaturalGrowth and Decay
  • Newton’s Law of Coolingthe cooling rate of a body isproportional to the differencebetween the temperature of thebody (T) and that of the surroundingmedium (M) DT = −k(T − M ) ! dt
  • Example - Newtons Law of Cooling
  • Wilhelmy’s LawThe velocity of a reaction is proportionalto difference between the initialconcentration of the reagent (a) and theamount transformed at any time (x) dx = k(a − x) 0 ≤ x ≤ a ! dt
  • Wilhelmy’s Law ExampleIn a particular reaction theconcentration of a reagent is10 (g) and 4 (g) have alreadybeen transformed. Find theamount left after 5 minutes
  • General ExampleBrine containing 2 g of salt per litre flowsinto a tank initially filled with 50 litres ofwater containing 10 g of salt. If the brineenters the tank at 5 litres /minute, theconcentration is kept uniform by stirring, andthe mixture flows out at the same rate, findthe amount of salt in the tank after 10minutes.