A similar study asked students to memorize 50 Hawaiian...

A similar study asked students to memorize 50 Hawaiian words, a language that is both unfamiliar and phonetically foreign to them (see Exercise 77 ). The average number of words retained is modeled by the function $W(t)=\frac{4 t+20}{t},$ where $W(t)$ represents the number of words after t days. a. Graph the function over the interval $t \in[0,40] .$ How many days until only half the words are remembered? How does this compare to Exercise $77 ?$ How many days until only one-fifth of the words are remembered? b. After 7 days, what is the average number of words retained? How many days until only 5 words can be recalled? c. What is the significance of the horizontal asymptote (what does it mean in this context)?

A similar study asked students to memorize 50 Hawaiian words, a language that is both unfamiliar and phonetically foreign to them (see Exercise 77 ). The average number of words retained is modeled by the function $W(t)=\frac{4 t+20}{t},$ where $W(t)$ represents the number of words after t days.
a. Graph the function over the interval $t \in[0,40] .$ How many days until only half the words are remembered? How does this compare to Exercise $77 ?$ How many days until only one-fifth of the words are remembered?
b. After 7 days, what is the average number of words retained? How many days until only 5 words can be recalled?
c. What is the significance of the horizontal asymptote (what does it mean in this context)?

Key Concepts

Equation Solving

Equation solving in the context of function models involves setting the function equal to a particular target value and solving for the variable of interest. This technique allows one to determine key moments or thresholds in a process, such as the point where a certain proportion of a quantity is retained or lost.

Graphing Functions

Graphing functions involves plotting the set of points defined by the function equation on a coordinate system. For rational functions, this includes identifying key features such as intercepts, asymptotes, and behavior at endpoints and discontinuities, which helps in visualizing trends and understanding underlying patterns in the data.

Modeling and Data Interpretation

Modeling and data interpretation involves using mathematical functions to represent real-world phenomena and then understanding and explaining what the components and behavior of these functions signify in practical terms. This concept bridges abstract mathematical relationships with tangible outcomes, such as the retention of learned information over time.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. These functions are notable for having restrictions in their domains where the denominator becomes zero and often exhibit interesting behaviors such as asymptotes. They are widely used in modeling scenarios where one quantity inversely relates to another, providing insights into dynamics that involve decay or growth processes.

Horizontal Asymptote

A horizontal asymptote represents a line that a function approaches as the independent variable heads towards infinity or negative infinity. In the context of modeling, it often indicates a limiting value the modeled variable will never truly drop below or rise above, reflecting long-term behavior or equilibrium conditions in real-world scenarios.

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