Acquisition of Language The percentage p(t) of children who...

Acquisition of Language The percentage $p(t)$ of children who can speak in at least single words by the age of $t$ months can be approximated by the equation $^{14}$ $p(t)=100\left(1-\frac{12,200}{t^{4.48}}\right) \quad(t \geq 8.5)$ a. Give a technology formula for $p$. b. Graph $p$ for $8.5 \leq t \leq 20$ and $0 \leq p \leq 100$. c. Create a table of values of $p$ for $t=9,10, \ldots, 20$ (rounding answers to one decimal place). d. What percentage of children can speak in at least single words by the age of 12 months? e. By what age are $90 \%$ or more children speaking in at least single words?

Acquisition of Language The percentage $p(t)$ of children who can speak in at least single words by the age of $t$ months can be approximated by the equation $^{14}$
$p(t)=100\left(1-\frac{12,200}{t^{4.48}}\right) \quad(t \geq 8.5)$ a. Give a technology formula for $p$.
b. Graph $p$ for $8.5 \leq t \leq 20$ and $0 \leq p \leq 100$.
c. Create a table of values of $p$ for $t=9,10, \ldots, 20$ (rounding answers to one decimal place).
d. What percentage of children can speak in at least single words by the age of 12 months?
e. By what age are $90 \%$ or more children speaking in at least single words?

Key Concepts

Interpreting Results in Context

Interpreting results in context ensures that the outcomes derived from mathematical operations are meaningful in the real-world scenario. It connects the abstract numerical results back to the situation being modeled, allowing for informed conclusions and decisions.

Solving Equations

Solving equations is a fundamental mathematical technique used to find the values of variables that satisfy a given condition. This is essential for determining critical points or thresholds within a model, such as identifying when a certain criterion is met.

Creating a Table of Values

The process of creating a table of values involves systematically computing the output of a function for a set of input values. This method is crucial for validating the behavior of a model, supporting graphical representations, and offering discrete data points for analysis.

Function Notation and Evaluation

Function notation is a formal way to express relationships between variables, where a function maps inputs to outputs. Evaluating the function at specific input values is important to understand the behavior of the process over its domain and to compute results for particular cases.

Mathematical Modeling

Mathematical modeling involves creating abstract representations of real-world phenomena using mathematical expressions. This allows the behavior of a process to be captured and analyzed through functions, providing insights and predictions based on changes in relevant parameters.

Graphing Functions

Graphing functions is a key concept that provides a visual representation of the relationship between variables. It helps in identifying trends, thresholds, and overall behavior of the model across a specified range, thereby facilitating better insight into the dynamics of the process being studied.

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