Depth of Snowfall Snow began falling at noon on Sunday. The...

Depth of Snowfall Snow began falling at noon on Sunday. The amount of snow on the ground at a certain location at time $t$ was given by the function $$h(t)=11.60 t-12.41 t^{2}+6.20 t^{3}$$ $$-1.58 t^{4}+0.20 t^{5}-0.01 t^{6}$$ where $t$ is measured in days from the start of the snowfall this function and use your graph to answer the following questions. (a) What happened shortly after noon on Tuesday? (b) Was there ever more than 5 in. of snow on the ground? If so, on what day(s)? (c) On what day and at what time (to the nearest hour) did the snow disappear completely?

Depth of Snowfall Snow began falling at noon on Sunday. The amount of snow on the ground at a certain location at time $t$ was given by the function
$$h(t)=11.60 t-12.41 t^{2}+6.20 t^{3}$$
$$-1.58 t^{4}+0.20 t^{5}-0.01 t^{6}$$
where $t$ is measured in days from the start of the snowfall this function and use your graph to answer the following questions.
(a) What happened shortly after noon on Tuesday?
(b) Was there ever more than 5 in. of snow on the ground? If so, on what day(s)?
(c) On what day and at what time (to the nearest hour) did the snow disappear completely?

Key Concepts

Mathematical Modeling

Mathematical modeling is the process of representing real-life phenomena using mathematical constructs such as functions or equations. By developing a model, one can simulate events like snowfall accumulation and melting over time and answer specific questions about the behavior of the phenomenon, bridging the gap between abstract mathematical analysis and practical interpretation.

Function Evaluation and Inequalities

Evaluating a function at specific points or solving inequalities involves determining where the function's values meet certain conditions, such as exceeding or falling below a threshold. This key concept is essential when assessing situations like whether the snow depth surpassed a particular measurement at given times, as it requires analyzing the function against set limits.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are important for modeling because they can approximate a wide variety of behaviors with continuous, smooth curves. In such contexts, higher degree polynomials can capture complex real-world phenomena by accounting for both rapid changes and gradual trends over time.

Graphical Analysis

Graphical analysis involves plotting a function to visually interpret its behavior, such as identifying intervals of increase or decrease, local extrema, inflection points, and zeros. This technique is particularly useful in understanding how the function behaves over a specific interval, such as tracking the progression of snowfall over time, and aids in answering questions about what the function’s graph reveals at certain times.

Roots of a Function

Finding the roots of a function means solving for the values of the variable where the function equals zero. In practical scenarios, these roots often correspond to critical events, such as the point when a certain measure (like snow depth) reaches zero. Recognizing and calculating these roots is crucial for determining the timing of events modeled by the polynomial.

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