Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Language dating Glottochronology is a method of dating a language at a particular stage, based on the theory that over a long period of time linguistic changes take place at a fairly constant rate. Suppose that a language originally had \(N_{0}\) basic words and that at time \(t\), measured in millennia ( 1 millennium \(=1000\) years), the number \(N(t)\) of basic words that remain in common use is given by \(N(t)=N_{0}(0.805)^{t}\) (a) Approximate the percentage of basic words lost every 100 years. (b) If \(N_{0}=200\), sketch the graph of \(N\) for \(0 \leq t \leq 5\).

Step-by-Step Solution

Answer

1.95% words are lost every 100 years. The graph shows exponential decay from 200 to about 68.1 over 5 millennia.

Step 1: Understanding the Equation

The function \(N(t)=N_{0}(0.805)^{t}\) represents the number of basic words that remain after \(t\) millennia, where \(N_{0}\) is the initial number of basic words. Each millennium, \(19.5\%\) of the words remain since \(0.805 = 1 - 0.195\). Thus, each millennia, \(19.5\%\) of the words are lost.

Step 2: Calculating Words Lost Per 100 Years

Since each millennium is 1000 years and words are reduced by \(19.5\%\) in this time, a linear approximation can be used to estimate words lost per 100 years. The fraction of words lost per 100 years is \(\frac{19.5\%}{10} = 1.95\%\). Thus, approximately \(1.95\%\) of words are lost every 100 years.

Step 3: Graphing \(N(t)\) for \(N_{0}=200\) Over 5 Millennia

Using \(N(t) = 200 \times (0.805)^{t}\), compute values of \(N(t)\) at \(t = 0, 1, 2, 3, 4, 5\). Evaluate: - \(N(0) = 200 \)- \(N(1) = 200 \times 0.805 \approx 161\)- \(N(2) = 200 \times (0.805)^2 \approx 129.8\)- \(N(3) = 200 \times (0.805)^3 \approx 104.6\)- \(N(4) = 200 \times (0.805)^4 \approx 84.4\)- \(N(5) = 200 \times (0.805)^5 \approx 68.1\)Plot these points on a graph with \(t\) from 0 to 5 and \(N(t)\) from 0 to 200. Observe the exponential decay.

Key Concepts

Linguistic Changes

Languages are constantly evolving. Over long periods, words naturally fall out of use while new words may be introduced. This continuous transformation is known as linguistic change. It's an inherent process fueled by factors such as cultural evolution, technological advances, and interactions with other languages. In glottochronology, these changes are used to estimate the "age" of a language. By monitoring the loss of basic words, linguists can date a language's stage of development. Glottochronology assumes that linguistic changes occur at a stable rate, akin to a linguistic "clock," making it possible to gauge time through these changes.

Exponential Decay

Exponential decay describes a process that decreases rapidly over time, usually at a constant percentage rate. This is represented mathematically by the equation \[ N(t) = N_0 (0.805)^t \] where \(N(t)\) is the quantity remaining after time \(t\), and \(N_0\) is the initial quantity. In our case, this equation details how the number of basic words in a language decays over many millennia. Each millennium, 19.5% of the remaining basic words vanish. This diminishing rate creates a curve that decreases exponentially, showing a sharp declining trend as time progresses. Exponential decay is a common mathematical model in both natural and social sciences.

Percentage Loss

Percentage loss refers to the proportion of an initial quantity that is lost over a specified period. In glottochronology, percentage loss helps estimate how many basic words disappear from a language over time. Based on the exponential decay formula, the loss of basic words is calculated per millennium, showing a reduction of 19.5%. For finer time measurements, this can be broken down into smaller intervals, such as per century. By dividing the millennium percentage by 10, it's estimated that about 1.95% of basic words are lost every 100 years. This method of calculation provides a practical way to understand word loss over shorter periods, which might be more relatable in historical studies.

Graphing Exponential Functions

Graphing exponential functions allows us to visualize how quantities like basic word counts change over time. Using the specific function \[ N(t) = 200 \times (0.805)^t \] we can plot this exponential decay from \(t = 0\) to \(t = 5\) millennia. Calculating \(N(t)\) for each millennium (e.g., \(N(0) = 200\), \(N(1) \approx 161\), \(N(2) \approx 129.8\), etc.) and plotting these values, reveals a curve that descends rapidly at first and then levels out, showcasing the exponential nature of decay. Through graphing, the rate and nature of linguistic decay becomes visually clear, providing an intuitive understanding of how basic words diminish over extensive periods of time.