Problem 95 - Algebra and Trigonometry

Question

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?

Step-by-Step Solution

Answer

(a) Snow stopped accumulating shortly after noon on Tuesday; (b) Snow exceeded 5 inches on Tuesday; (c) Snow disappeared completely on Friday afternoon.

Step 1: Understanding the Function's Components

We are given a polynomial function \( h(t) = 11.60t - 12.41 t^2 + 6.20 t^3 - 1.58 t^4 + 0.20 t^5 - 0.01 t^6 \) that represents the depth of snowfall in inches as a function of time \( t \), where \( t \) is measured in days from noon on Sunday.

Step 2: Calculate Key Values via Derivatives

To understand what happened shortly after noon on Tuesday, we need to analyze the critical points of the function, which are found by taking the derivative and solving \( h'(t) = 0 \). This will provide the times when snowfall rate changes or stops.

Step 3: Evaluate the Function for 4.5 Days

Since noon on Sunday is \( t = 0 \), noon on Tuesday is \( t = 2 \), and shortly after noon is \( t > 2 \). We examine \( t \) around \( 2.5 \) and \( 3 \) (Monday noon to Wednesday noon) using the derivative \( h'(t) \). Find \( h(2.5), h(3) \), and other values to observe trends.

Step 4: Checking for Height Greater than 5 Inches

To determine if the snow height was ever more than 5 inches, evaluate the function \( h(t) \) over the domain periodically, specifically around key points such as maximum values. Check over 1-day intervals to identify when this threshold is exceeded.

Step 5: Determine When Snow Disappears Completely

The snow is completely gone when \( h(t) = 0 \). Solve this equation over the range of reasonable days after the start — typically by using numerical methods or graphing techniques to find when the function first returns to zero after any growth.

Step 6: Analyze Results for Each Question

(a) After noon on Tuesday, if \( h'(t) \) shows a shift from positive to zero or negative, it indicates key transitions such as peak or end of snow accumulation. (b) Intervals where \( h(t) > 5 \) are noted days after snowfall starts. (c) Solve \( h(t) = 0 \) for time \( t \) to the nearest hour, indicating when the snow depth becomes zero.

Step 7: Interpret the Results

Based on derivative trends and calculations, interpret when the snowfall started receding or disappeared completely, and identify exact time slices where the snowfall exceeded any thresholds, such as the 5 inch mark.

Key Concepts

Understanding Derivatives and Their Role

When working with polynomial functions, especially in applications like snow depth calculation, derivatives are incredibly useful. They allow us to understand the behavior of the function over time. The derivative of a function, notated as \( h'(t) \) in this case, represents the rate of change of the function. It tells us how fast the snow depth is increasing or decreasing at any given time.

• To find critical points, where the rate of change shifts, you set the derivative equal to zero: \( h'(t) = 0 \).
• This helps identify moments when the snow stops gathering or starts melting faster than it accumulates.

By analyzing derivatives, we understand day-to-day changes in snow depth, and estimate when major shifts, like peaks or full disappearances, occur.

Finding Critical Points

Critical points are key in understanding the overall trend of snow accumulation or melting. These points occur where the derivative \( h'(t) \) becomes zero or undefined, signaling a change in the slope of the graph. These changes could indicate a peak or a trough in the snow accretion pattern.
To identify these:

• Compute the derivative of the snowfall function \( h(t) \).
• Solve \( h'(t) = 0 \) to find the values of \( t \) where snow either stops increasing or starts decreasing.

By determining critical points, you gain insight into the times after a heavy snowfall or when major melting begins.

Snow Depth Calculation Over Time

Calculating snow depth over time involves substituting various time values into the polynomial function \( h(t) \). This determines how much snow is on the ground at specific moments. For example, during step-by-step snow analysis, you check snow depth at unique times like noon on Tuesday (\( t = 2\)) and a couple of hours after (e.g., \( t = 2.5 \)).

• Insert different \( t \) values to the function to find when critical points occur, as well as snow quantity transitions.
• You can predict if snow exceeds a certain depth, like 5 inches, by checking if \( h(t) > 5 \).

Understanding this helps in predicting severe weather impacts, allowing for better preparation during winters.

Using Graphing Techniques

Graphing is an effective way to visualize polynomial functions like the snowfall equation \( h(t) \). When you graph this function, it brings clarity to how snowfall and melting change over the time frame. You can easily spot critical points and transitions visually.

• Plot points for different values of \( t \) to see peaks and troughs.
• Trace over intervals to pinpoint precisely when the depth of snow increases, decreases, or disappears altogether.

Graphing provides an intuitive grasp of the problem, making it easier to interpret complex data without solving equations exclusively by calculation.