Snow began falling at noon on Sunday. The amount of snow on...

Snow began falling at noon on Sunday. The amount of snow on the eground 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 and $h(t)$ is the depth of snow in inches. Draw a graph of 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?

Snow began falling at noon on Sunday. The amount of snow on the eground 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 and $h(t)$ is the depth of snow in inches. Draw a graph of 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

Interpreting Mathematical Models in Context

Interpreting mathematical models involves translating the behavior of a function into meaningful insights about real-world phenomena. This means understanding the significance of the input and output units, recognizing when events like accumulation, peak, or depletion occur, and using the graph to compare these events to the timeline in the scenario, such as pinpointing time-specific observations like the occurrence just after noon on Tuesday.

Finding Zeros of Functions

Finding the zeros (roots) of a function, where the output equals zero, is important in practical applications because these correspond to times when an event, like the snow completely melting, occurs. In the context of this problem, solving for when the snow depth is zero helps identify the exact moment when the snowfall has ceased to have any impact.

Critical Points and Extrema

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or inflection points. Identifying these points is key for understanding when a function, such as the snow depth model, reaches a maximum accumulation of snow or when it begins to decrease, which is crucial for interpreting changes occurring shortly after specific times, like events occurring shortly after noon on Tuesday.

Polynomial Functions

Polynomial functions are expressions that involve only nonnegative integer powers of the variable. They are smooth, continuous, and differentiable, which means they can model a wide variety of real-world phenomena, including physical processes that change over time. Understanding their structure helps in analyzing how components like constant, linear, quadratic, and higher degree terms influence the overall behavior of the function.

Graphing Functions

Graphing a function involves plotting the function's output against its input to visualize the behavior over a prescribed domain. For polynomial functions, graphing can reveal important features such as intercepts, turning points, and end behavior. This visual understanding aids in answering questions about when certain events occur, such as when the value reaches a peak or returns to zero.

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