#### 432**Question:** A linguist is studying the evolution of language through a model where the frequency of a certain linguistic feature in a population is given by the function \( f(t) = 3t^2 - 2t + 5 \). Find the time \( t \) at which the frequency is 20.

["Understanding Language Evolution: Solving for When Linguistic Feature Frequency Reaches 20", "The dynamic nature of language offers a fascinating lens through which linguists study how linguistic features spread and evolve across populations. One powerful approach involves modeling the frequency of a specific linguistic trait over time using mathematical functions. In this article, we explore a real-world application where a linguist uses a quadratic model to analyze language change and solve a key question: At what time t does the frequency of a linguistic feature reach exactly 20?", "### The Mathematical Model of Linguistic Frequency", "Consider the function that describes the frequency ( f(t) ) of a linguistic feature at time ( t ):\n[ f(t) = 3t^2 - 2t + 5 ]\nHere, ( f(t) ) represents the observed frequency of a particular grammatical or phonological pattern, such as the use of a specific syntactic construction, dialectal pronunciation, or lexical item. This quadratic model captures how linguistic behavior accumulates or shifts within a speech community—often reflecting social, cognitive, and communicative pressures.", "### The Core Question: When Does Frequency Equal 20?", "Our focus is on solving the equation:\n[ 3t^2 - 2t + 5 = 20 ]\nWe seek the time t (in relevant units—years, generations, or linguistic cycles—depending on context) when the feature reaches a frequency threshold of 20, a benchmark often used to assess linguistic salience or sociolinguistic impact.", "### Step 1: Set up the equation\nSubtract 20 from both sides:\n[ 3t^2 - 2t + 5 - 20 = 0 ]\n[ 3t^2 - 2t - 15 = 0 ]", "### Step 2: Apply the quadratic formula\nThis is a standard quadratic equation of the form ( at^2 + bt + c = 0 ), with coefficients:\n- ( a = 3 )\n- ( b = -2 )\n- ( c = -15 )", "The quadratic formula gives solutions:\n[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]\nCompute the discriminant:\n[ \Delta = (-2)^2 - 4(3)(-15) = 4 + 180 = 184 ]", "Now substitute:\n[ t = \frac{-(-2) \pm \sqrt{184}}{2 \cdot 3} = \frac{2 \pm \sqrt{184}}{6} ]", "Simplify ( \sqrt{184} ):\n[ \sqrt{184} = \sqrt{4 \cdot 46} = 2\sqrt{46} ]\nThus,\n[ t = \frac{2 \pm 2\sqrt{46}}{6} = \frac{1 \pm \sqrt{46}}{3} ]", "### Step 3: Interpret the solutions", "Since ( t ) represents time, we consider only real, positive values. Both roots are real:\n- ( t_+ = \frac{1 + \sqrt{46}}{3} \approx \frac{1 + 6.782}{3} \approx \frac{7.782}{3} \approx 2.594 )\n- ( t_- = \frac{1 - \sqrt{46}}{3} \approx \frac{1 - 6.782}{3} \approx \frac{-5.782}{3} \approx -1.927 ) (discard, negative time)", "### Final Answer\nThe frequency of the linguistic feature reaches 20 at approximately:\n[ \boxed{t = \frac{1 + \sqrt{46}}{3}} \approx 2.594 ] units (time, generation, or cycle count, depending on model calibration).", "### Why This Matters in Linguistics", "This solution enables linguists to pinpoint critical transition points—when a linguistic innovation crosses a frequency threshold that may signal adoption, stabilization, or resistance. It bridges quantitative modeling with qualitative interpretation, illuminating patterns behind language change, dialect contact, and cultural transmission.", "For researchers and enthusiasts alike, modeling frequency dynamics transforms abstract models into testable predictions—revealing how language evolves not in isolation, but as a measurable, evolving phenomenon in human societies.", "---", "Keywords: linguistic evolution, language frequency modeling, quadratic function, frequency threshold, computational linguistics, sociolinguistic modeling, time-evolution of language, ( f(t) = 3t^2 - 2t + 5 ), solving for ( t ) in language models."]