Some Applications Involving Separable Differential Equations Mixing Problem

Some Applications Involving Separable Differential Equations

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?

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?

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 C 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.

Radioactive Decay The half-life of radium-226 is 1590 years. (A) 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. (B) Find the mass after 1000 years. (C) When will the mass be reduced to 30 mg?

Newton’s Law of Cooling A thermometer is taken from a room where the temperature is 200 C to the outdoors where the temperature is 50 C. After one minute, the temperature reads 120 C. (A) What will the reading on thermometer be after one more minute? (B) At what time will thermometer read 60 C?