There are four squirrels currently living in Linda's attic....

There are four squirrels currently living in Linda's attic. If she does nothing to evict these squirrels, the number of squirrels in her attic after $t$ days will be given by the formula $S(t)=\frac{12+5.5 t}{3+0.25 t} .$ (a) Verify that there are four squirrels in Linda's attic at time $t=0$. (b) Determine the number of squirrels in Linda's attic after 30 days, 60 days, and one year. (c) Approximate $\lim _{t \rightarrow \infty} S(t)$ with a table of values. What does this limit mean in real-world terms? (d) Graph $S(t)$ with a graphing utility, and use the graph to verify your answer to part (c).

There are four squirrels currently living in Linda's attic. If she does nothing to evict these squirrels, the number of squirrels in her attic after $t$ days will be given by the formula $S(t)=\frac{12+5.5 t}{3+0.25 t} .$
(a) Verify that there are four squirrels in Linda's attic at time $t=0$.
(b) Determine the number of squirrels in Linda's attic after 30 days, 60 days, and one year.
(c) Approximate $\lim _{t \rightarrow \infty} S(t)$ with a table of values. What does this limit mean in real-world terms?
(d) Graph $S(t)$ with a graphing utility, and use the graph to verify your answer to part (c).

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Key Concepts

Evaluation of Functions

This concept involves substituting a specific value into a function to determine its output. When given a mathematical formula, such as those representing real-world phenomena, evaluating the function at particular instants (like t=0) is essential to verify initial conditions or specific predictions.

Rational Functions

Rational functions are ratios of two polynomials and are common in mathematical models of real-world systems. They exhibit properties such as vertical and horizontal asymptotes, which help in understanding the function's behavior both at specific points and in the long-term as the variable grows large.

Limits and End Behavior

The limit of a function, especially as the variable approaches infinity, provides insight into the function's long-term behavior. For rational functions, this typically involves finding a horizontal asymptote, which indicates the value the function approaches, revealing how systems stabilize or plateau over time.

Graphing Functions for Verification

Graphing a function is a visual method to confirm analytic results, such as calculating limits or initial values. A visual representation also aids in understanding the overall behavior of the function, including trends, asymptotic behavior, and points of intersection with axes.

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