Greenwood press
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book-20600
days
start of day end of day 1 200.000 120.000 2 320.000 192.000 3 392.000 235.200 4 435.200 261.120 5 461.120 276.672 6 476.672 286.003 days start of day end of day 7 486.003 291.602 8 491.602 294.961 9 494.961 296.977 10 496.977 298.186 11 498.186 298.912 12 498.912 299.347 13 499.347 299.608 14 499.608 299.765 15 499.765 299.859 16 499.859 299.915 17 499.915 299.949 18 499.949 299.970 19 499.970 299.982 20 499.982 299.989 The amount of drug (in milligrams) in a person’s bloodstream when 60 percent remains from the previous day and an additional 200 mg are added each day. A pharmacist can modify this initial amount on the first day and observe changes in the limit of this sum to determine that 80 mg is an appropriate daily dosage to maintain 200 mg in the bloodstream over time, as shown below. days start of day end of day 1 80.000 48.000 2 128.000 76.800 3 156.800 94.080 4 174.080 104.448 5 184.448 110.669 6 190.669 114.402 7 194.401 116.641 8 196.641 117.984 9 197.984 118.791 10 198.791 119.274 11 199.274 119.565 12 199.565 119.739 13 199.739 119.843 118 SERIES days start of day end of day 14 199.843 119.906 15 199.906 119.944 16 199.944 119.966 17 199.966 119.980 18 199.980 119.988 19 199.988 119.993 20 199.993 119.996 The amount of drug (in milligrams) in a person’s bloodstream when 60 percent remains from the previous day and an additional 80 mg are added each day. This situation is an example of a geometric series, since the amount remaining in the bloodstream is affected by a constant ratio of 60 percent. The sum can be re- written as days since last dosage 1 2 3 4 80 + 80(0.60) 1 + 80(0.60) 2 + 80(0.60) 3 + 80(0.60) 4 + . . . The sum, s, can be determined by the equation s = g 1 (1−r n ) 1−r , where g 1 is the initial dosage, r is the constant ratio, and n is the number of days the dosage is taken. Since the number of days that the drug is taken is unknown, pharmacists need to examine situations in which the drug is taken indefinitely. Therefore the sum of an infinite geometric series is s = g 1 1−r because lim n→∞ g 1 (1−r n ) 1−r = g 1 1−r when |r| < 1. In this case, the desired sum, s, is 200 mg, r is 60 percent or 0.60, and g 1 is unknown. Substituting the values into the equation, you will get 200 = g 1 1−0.60 , and a solution of g 1 = 80 mg. Thus the doctor needs to make pre- scriptions of 80 mg each day in order to maintain the desired dosage of 200 mg. Geometric series are also used to predict the amount of lumber that can be cut down each year in a forest to ensure that the number of trees remain at a sta- ble level. Each year, forest rangers plant seeds for new trees to account for those chopped down and lost to forest fires. Suppose the ranger wants to know what proportion of trees they can afford to lose or remove each year if they plant 500 new trees and want to consistently maintain 80,000 trees in the forest. After sub- stituting s and g 1 in the formula s = g 1 1−r , the unknown value for r is 0.00625. This means that the forest ranger wants to maintain 99.375 percent of the trees each year. However, an interesting phenomenon is to notice that the forest can recover from a disaster such as a fire in a reasonably short period of time. Suppose a fire destroys 35 percent of the trees in the forest, leaving 52,000 trees. If 500 new trees are planted each year, and 0.625 percent of the total number of Download 1.81 Mb. Do'stlaringiz bilan baham: |
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