Question

Over time, the number of original basic words in a language tends to decrease as words become obsolete or are replaced with new words. Linguists have used calculus to study this phenomenon and have developed a methodology for dating a language, called glottochronology. Experiments have indicated that a good estimate of the number of words that remain in use at a given time is given by $N(t)=N_{0} e^{-0.217 t}$, where N(t) is the number of words in a particular language, t s measured in the number of millennium, and $N_0$ is the original number of words in the language. a. In 1950, C. Feng and M. Swadesh established that of the original 210 basic ancient Chinese words from 950 A.D., 167 were still being used. Letting t=0 corresponds to 950, with $N_0$=210, find the number of words predicted to have been in use in 1950 A.D., and compare it with the actual number in use. b. Estimate the number of words that will remain in the year 2050 (t=1.1). c. Find N'(1.1) and interpret your answer.

Solution

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As indicated by experiments, a good estimate of the number of words that remain in use at a given time is given by

$N(t) = N_{0}e^{-0.217t},$

where $N(t)$ is the number of words in a particular language ,$t$ is measured in the number of millenium and $N_{0}$ is the original number of words in the language.

$\textbf{Answer for (a) }$ We have to find the number of words predicted to have been in use in 1950 A. D., and compare it with the actual number in use. Here, $N(t) = N_{0}e^{-0.217t},$ where $t = 1$ and $N_{0} = 210$.

\begin{aligned} N(1) &= 210e^{-0.217\times 1}\\ &\approx 169. \end{aligned}

Hence the number of words predicted to have been in use in 1950 A.D is 169, and the actual number in use was 167.

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