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

Acquisition of Language The percentage $p(t)$ of children who can speak in sentences of five or more words by the age of $t$ months can be approximated by the equation ${ }^{15}$ $$p(t)=100\left(1-\frac{5.27 \times 10^{17}}{t^{12}}\right) \quad(t \geq 30)$$ a. Give a technology formula for $p$. b. Graph $p$ for $30 \leq t \leq 45$ and $0 \leq p \leq 100$. c. Create a table of values of $p$ for $t=30,31, \ldots, 40$ (rounding answers to one decimal place). d. What percentage of children can speak in sentences of five or more words by the age of 36 months? e. By what age are $75 \%$ or more children speaking in sentences of five or more words?

Acquisition of Language The percentage $p(t)$ of children who can speak in sentences of five or more words by the age of $t$ months can be approximated by the equation ${ }^{15}$
$$p(t)=100\left(1-\frac{5.27 \times 10^{17}}{t^{12}}\right) \quad(t \geq 30)$$
a. Give a technology formula for $p$.
b. Graph $p$ for $30 \leq t \leq 45$ and $0 \leq p \leq 100$.
c. Create a table of values of $p$ for $t=30,31, \ldots, 40$ (rounding answers to one decimal place).
d. What percentage of children can speak in sentences of five or more words by the age of 36 months?
e. By what age are $75 \%$ or more children speaking in sentences of five or more words?

Key Concepts

Mathematical Modeling

Mathematical modeling involves representing real-world phenomena using mathematical formulas or equations. In contexts like developmental studies, a model is used to approximate how a certain ability develops over time, allowing predictions and analyses based on specific inputs.

Function Evaluation

Function evaluation is the process of substituting a value for the independent variable in a function to determine the corresponding output. This concept is crucial when calculating specific outcomes for given inputs, such as determining a percentage at a certain age.

Graphing Functions

Graphing functions is the visual representation of the relationship between variables. By plotting the function on a coordinate system, one can observe trends, patterns, and behaviors, which is particularly useful for understanding how a modeled phenomenon changes over time.

Creating a Table of Values

Creating a table of values involves computing and organizing the outputs of a function for a range of specified inputs. This step-by-step method helps verify the behavior of the function and provides a clear, numerical representation of the relationship between the variables.

Solving Equations for Specific Outputs

Solving equations for specific outputs involves finding the input value that results in a particular output from the function. This inverse problem is essential for determining when a certain threshold is met, such as when a modeled percentage reaches a predetermined level.

Domain and Range Considerations

Domain and range considerations are fundamental in understanding the limitations and applicable values of a function. The domain specifies the possible inputs (such as age in months), and the range indicates the possible outputs (such as percentages), ensuring the model is used appropriately within its intended context.

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