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Some Applications Involving Separable Differential Equations

Some Applications Involving Separable Differential Equations. Mixing Problem.

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Some Applications Involving Separable Differential Equations

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  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. Newton’s 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?

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