24 THE SYSTEMS CONCEPT 



which, if reflected upon, will become quite obvious because it is not only 

 a "natural" law, an observed law of Nature, but also a logical deduction. 



In our examples, most commonly a decay is involved, in this case the 

 decay of a concentration. Thus k is a negative number. If the minus sign 

 is taken out of the k and k replaced by — X, the expression becomes N = 

 N e~ Xl , sometimes written N = N exp(-Xt), for radioactive decay, where 

 JV is the number of particles present when t = 0. 



Figure 1-8 shows the shape of the exponential curve for positive k values 

 (growth), and for negative k values (decay). Note that the former increases 

 to infinity, unless checked by the onset of some other law; and that the latter 

 decays toward zero, reaching zero only after an infinitely long time, although 

 it may be below the lowest measureable value within a very short time. The 

 larger the value of k, the faster the growth curve sweeps upwards, and the 

 sooner the decay curve approaches zero. 



PROBLEMS 



1-1 : (a) If a student must pass biochemistry, and John is a student, then . . . ? 



(b) If y = 2x and Z = y, then what functional relationship exists between 

 Z and*? 



(c) Uy =/,(*) and £ = f 2 (x); and f 2 (x) = /, (x) -f 3 (x), then what is the rela- 

 tionship between x andy? 



(d) If A °c B, and B °c C, what is the relationship between A and C? 



(e) If the weight of a given volume of gas is proportional to density, and if the 

 density is proportional to its pressure, then what is the relationship between 

 weight of a given volume and its pressure? 



1-2: Choose at random, alphabetically for example, the heights in inches of 

 25 students. 



(a) Is the distribution normal? Was the sample biased? 



(b) What are the average deviation, Ax, and the standard deviation, a? 



(c ) What fraction of the sample falls within the mean deviation from the mean? 



(d) What fraction of the sample falls within one standard deviation from the 

 mean? If the distribution had been normal, what would have been the 

 fraction? 



(e) What fractions of the sample fall with ±2 a and ±3 a? If the distribution 

 had been normal, what would have been the fractions? 



1-3: Make a table showing how the distance fallen, the speed, and the acceleration 

 of a parachutist change in the first 5 sec before the chute opens. (Make the 

 calculations for each second.) 



Suppose he hits the earth at a velocity of 120 ft per sec without the chute 

 opening. From what height did he jump? 



1-4: The decay of Sr 90 follows the exponential law N = JV e~ Xl , where N is the 

 concentration of radiating material at any time, t; N Q is the concentration at 

 some arbitrary zero of time; and X is the decay constant of Sr 90 , namely 0.028 

 years" 1 (i.e., 0.028 is the fraction lost per year). 



