\section*{Exercise 4 - 081234}

Through the improvement of living conditions and health protection, tuberculosis could be successfully combated in the GDR.

While in 1950 there were still 92,760 cases of active tuberculosis, this number decreased to 13,777 by 1966 over the following 16 years.

a) By what percentage did the number of cases decrease annually, assuming a constant annual percentage decrease (which, apart from minor fluctuations, corresponds to reality)?

b) How many years was the so-called half-life in the period from 1950 to 1966, i.e., the time it took for the number of cases to be reduced by half (answer in years with one decimal place)?

c) How many cases of infection can be expected in 1970 if a constant annual percentage decrease is assumed?
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a) Let \(q\) be the (assumed constant) ratio of the number of disease cases in one year compared to the previous year.

According to the problem statement, \(92760 \cdot q^{16} = 13777\) or \(q = \sqrt[16]{\frac{13777}{92760}}\). Using a calculator, one obtains \(q \approx 0.8876\), meaning that the number of disease cases has decreased by approximately \(11.24\%\) per year on average.

Remark: When this problem was posed, calculators were not yet available, but logarithm tables were.

From these, one can easily read \(\lg (1.3777) \approx 0.139\) and \(\lg (9.2670) \approx 0.967\), from which \(\lg \left(\frac{13777}{92760}\right) \approx 0.139 - 0.967 = -0.828\) and thus \(\lg (q) \approx -\frac{0.828}{16} \approx -0.052\) is obtained. Again, one can read from such a table the value \(10^{1-0.052} = 10^{0.948} \approx 8.872\), leading to \(q \approx 0.8872\). The calculation naturally becomes more accurate if not only three-digit mantissas, as assumed here, are available. The question also arises for which values the tables were available...

b) For the half-life \(t\), measured in years, \(q^{t} = \frac{1}{2}\) holds, so \(t \cdot \lg (q) = -\lg (2)\) or \(t = -\frac{\lg (2)}{\lg (q)} \approx 5.8\), meaning that the number of disease cases is halved every 5.8 years.

c) If one assumes a constant percentage reduction in the number of disease cases per year, then from 1966 to 1970, the number of cases will decrease by a factor of \(q^{4}\), so one can expect approximately 13777 \cdot 0.8876^{4} \approx 8500 disease cases.