Evolution of Language The number of years, n, since two...

Evolution of Language The number of years, $n$, since two independently evolving languages split off from a common ancestral language is approximated by $$ n \approx-7600 \log r $$ where $r$ is the proportion of words from the ancestral language common to both languages. (a) Find $n$ if $r=0.9$ (b) Find $n$ if $r=0.3$ (c) How many years have elapsed since the split if half of the words of the ancestral language are common to both languages?

Evolution of Language The number of years, $n$, since two independently evolving languages split off from a common ancestral language is approximated by
$$
n \approx-7600 \log r
$$
where $r$ is the proportion of words from the ancestral language common to both languages.
(a) Find $n$ if $r=0.9$
(b) Find $n$ if $r=0.3$
(c) How many years have elapsed since the split if half of the words of the ancestral language are common to both languages?

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Key Concepts

Mathematical Modeling

This concept involves using mathematical formulas to represent real-world phenomena. In the context of language evolution, it refers to constructing a model that quantifies how the proportion of shared vocabulary between languages decreases over time, thereby allowing the estimation of elapsed time since divergence.

Logarithmic Functions

Logarithms are the inverses of exponential functions and are useful for transforming multiplicative relationships into additive ones. In this model, a logarithmic function is used to relate the proportion of shared words to the number of years since two languages diverged.

Exponential Decay

Exponential decay describes processes where a quantity decreases at a rate proportional to its current value. This concept underpins the model, as the shared vocabulary between languages decays exponentially over time, allowing the use of a logarithmic transformation to estimate the time elapsed since the split.

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