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 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?

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
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

Conversion and Interpretation of Time Measurements

Often, a function models time in continuous units (like days), but practical interpretation may require converting these values into more immediately understandable units such as hours or specific days of the week. This conversion is essential for accurately relating the mathematical analysis to real-life timing and events.

Interpreting Function Behavior in Context

Interpreting a function's behavior in a real-world context involves correlating mathematical features—such as increasing or decreasing trends, peaks, and zeros—with events or changes in the observed situation. This approach is fundamental in applied mathematics when using models to predict and explain phenomena.

Finding Zeros of Functions

Finding the zeros (or roots) of a function refers to determining the points at which the function's value is zero. In applied contexts, these zeros can represent key moments such as the start or end of an event. Solving polynomial equations to obtain these zeros is crucial for identifying when a process has started, ended, or reached a significant transition.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting the function over a chosen domain to visualize key behavior such as end behavior, turning points, and intercepts. This process helps in understanding how the function's output changes over time and identifying important events like peaks and zeros, which are essential for analyzing real-world phenomena modeled by the function.

Analyzing Local Extrema

Local extrema are the maximum or minimum values of a function within a particular interval. In real-world modeling, identifying these extrema helps in determining critical points such as the time at which an accumulation is at its highest. This analysis is important for understanding the peak behavior of the modeled phenomenon and its subsequent changes.

Transcript

00:01 Hello everyone, the question is snow begins falling at noon on sunday, the amount of the snow on the ground.

00:07 At a certain location at time t was given by the function.

00:12 The function is h of t.

00:19 This is height of the snow which is made in inches.

00:23 H of t equal to 11 .6 t minus 12 .4 t squared plus 6 .2 t cube minus 1 .58 t -dase to the per 4 plus 0 .02 t -dase to the power 5 minus 0 .01 t -dase to the per 6 where t is the time and it's measured in days so first of all what we have to find is what happens shortly after noon on tuesday so snow start started on noon on sunday from noon on sunday till noon on monday is one day.

01:13 From noon on monday till noon on tuesday as second day.

01:17 And we have to find what happens after t equal to two, shortly after t equal to two.

01:23 First of all, let's just plot this graph and see the behavior of it.

01:30 11 .6.

01:33 I'm using variable x instead of t here, but it doesn't change as anything.

01:39 12 .4x square, 6 .2x cube, and negative 1 .58 x raised to the power 4, 0 .2 plus 0 .2x raised to the bar 5, minus 0 .01 x raised to the power 5, minus 0 .01 x raised to the power 5.

02:43 So this is our graph and we have to see what happens shortly after t equal to 2.

02:53 Let's zoom it in.

03:01 This is t equal to 2.

03:03 Let's look at the value.

03:05 T equal to 2 you can see that our value of h height of snow is same.

03:13 When it's below 2 and when we go above 2 it stays same for some time.

03:28 We have magnified it a lot...