An industrial spill of hazardous materials has occurred, and your plant safety team has entered the building to begin the cleanup of the toxic substance. the team members are wearing protective suits to prevent the possible fatal side effects of prolonged exposure to the substance. the suits are constructed of a material that chemically neutralizes toxic substance. however, over time, the chemical coating in the garment's fibers breaks down and loses its ability to protect the wearer from exposure to the hazardous materials, experience shows that the cloth can neutralize 30 percent of the chemical it encounters per hour. experience also shows that, during periods of exposure, the garment encounters 12 micrograms of the chemical per hour. standards for human exposure to the toxic substance indicates that such suits are safe for use until they reach a level of 35 micrograms of the toxic chemical in their fibers. question: under these conditions, how many hours can the safety team work without having to worry about changing their outfits? this problem is easily modeled through the use of a discrete dynamical system. if we think of the time intervals, in hours, as the variable n and the amount of chemical in the suit at the end of an interval of length n hours as an, we can model the chemical residue in the suits by the relationship: an = 0.7 an-1 + 12 for n > 0 and ao = 0. here the 0.7 coefficient describes the amount of residue that remains in the suit after each period of one hour, -the treated garment cloth has neutralized the other 30 percent. the term 12 that is added reflects the new contamination. as this process starts from a clean suit, the initial value, ao is 0.