Problem 101 - Precalculus Mathematics for Calculus

Question

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 $$ \begin{aligned} h(t)=11.60 t &-12.41 t^{2}+6.20 t^{3} \\ &-1.58 t^{4}+0.20 t^{5}-0.01 t^{6}\end{aligned}$$ 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?

Step-by-Step Solution

Answer

(a) Snow depth decreased. (b) Yes, on Monday. (c) Snow disappeared on Wednesday noon (72 hours).

Step 1: Understanding the Function

The function given, \( h(t) = 11.60t - 12.41t^2 + 6.20t^3 - 1.58t^4 + 0.20t^5 - 0.01t^6 \), represents the snow depth in inches at time \( t \), measured in days since noon on Sunday. We need to analyze this function to understand the depth of snow over time.

Step 2: Sketching the Graph

To graph the function, calculate the depth of snow for a range of \( t \) values, from 0 to a few days after Sunday. Use graphing software or calculate points manually to get an idea of how snow depth changes over time. Note key features like maximum points and times when \( h(t) = 0 \).

Step 3: Analyzing Tuesday's Event

From the graph, examine what happens to the function \( h(t) \) after noon on Tuesday (where \( t = 2 \)). Look for significant changes, such as increases or decreases in snow depth, or when the snow depth becomes zero.

Step 4: Checking for 5 Inches of Snow

Identify intervals where \( h(t) > 5 \). This requires solving for \( t \) when \( h(t) = 5 \) and checking values in these intervals to confirm if and when more than 5 inches of snow was on the ground.

Step 5: Determining when Snow Disappears

Find the values of \( t \) where \( h(t) = 0 \). This will show when the snow completely disappeared. Check for solutions beyond noon on Sunday, as we are interested in subsequent days.

Step 6: Answer Questions with Insights

Based on the analysis and the graph: (a) Determine the snow depth shortly after noon on Tuesday. (b) Identify days when \( h(t) > 5 \). (c) Exact time \( t \) when \( h(t) = 0 \) to the nearest hour.

Key Concepts

Depth of Snow

The depth of snow is a key consideration in understanding environmental changes during snowy conditions. In our exercise, the depth of snow is represented by a polynomial function, which gives us the amount of snow on the ground over time in inches. Here, the variable \( t \) indicates time in days measured from the start of the snowfall at noon on Sunday.
Analyzing snow depth helps determine various weather-related aspects:

• Maximum snow depth and when it occurs
• Periods of accumulation versus melting
• When the snow completely melts away

This specific polynomial function is useful to predict and plan during snow events — for example, knowing when roads might need plowing or when it is safer to travel.

Graph Sketching

Sketching a graph of a polynomial function such as this snow depth function provides visual insight into how the depth changes over time. To do this, you must plot values of \( h(t) \) for various values of \( t \). This helps uncover patterns, trends, and key features of the function.
When sketching:

• Start by computing several key points, including starting and ending points.
• Identify when the function hits notable points, such as maximum depths or where it equals zero.
• Use graphing tools or software for accuracy, especially for more complex polynomials.

Sketching a graph not only aids in understanding the function's behavior but also in answering related questions such as changes on specific days.

Function Analysis

Function analysis involves examining the properties and behavior of the polynomial function itself. For this exercise, you are tasked with understanding how the snow depth changes over time. When analyzing the function:

• Determine critical points where the first derivative is zero, which could indicate maximum or minimum snow depth.
• Calculate where the function equals certain values (e.g., 5 inches) to find periods of significant snow cover.
• Solve for times when the snow depth returns to zero, indicating complete snow melt.

Through function analysis, deeper insights into the timing and extent of snowfall and melt can be gained.

Real-World Applications

Polynomial functions modeling natural phenomena like snowfall are more than theoretical exercises; they have real-world applications. Understanding how to read and apply these models has practical implications. These functions can be used for:

• Predicting weather patterns and planning for travel or transportation
• Public safety by determining when and where snow removal might be necessary
• Environmental studies to assess climate changes over time

Real-world applications of polynomial functions remind us that math is not just numbers and equations — it's a vital tool for interpreting and adapting to our surroundings effectively. By mastering these concepts, students are better prepared to apply mathematical knowledge to real-world challenges.