Problem 88 - College Algebra with Modeling and Visualization

Question

The monthly average high temperature in degrees Fahrenheit at Daytona Beach, Florida, can be approximated by \(f(x)=0.0145 x^{4}-0.426 x^{3}+3.53 x^{2}-6.22 x+72\) where \(x=1\) corresponds to January, \(x=2\) to February, and so on. Estimate graphically when the monthly average high temperature is \(75^{\circ} \mathrm{F}\) or more.

Step-by-Step Solution

Answer

Graph the equation, find intersections near March and November. Temperature exceeds 75°F from about March to November.

Step 1: Understand the problem

We are given the function \( f(x) = 0.0145x^4 - 0.426x^3 + 3.53x^2 - 6.22x + 72 \), which models the average high temperatures for each month at Daytona Beach. Our task is to find when the temperature is equal to or exceeds 75°F, using a graphical approach.

Step 2: Set up the equation for graphing

We need to find the x-values where the temperature \( f(x) \) is equal to or greater than 75°F. Set up the equation \( f(x) = 75 \). This gives us: \[ 0.0145x^4 - 0.426x^3 + 3.53x^2 - 6.22x + 72 = 75 \].

Step 3: Convert equation to find intersection

Subtract 75 from both sides of the equation to find the intersection points:\[ 0.0145x^4 - 0.426x^3 + 3.53x^2 - 6.22x - 3 = 0 \]. This function represents when the temperature is precisely 75°F.

Step 4: Graph the function

Graph \( g(x) = 0.0145x^4 - 0.426x^3 + 3.53x^2 - 6.22x - 3 \). The points where this graph intersects the x-axis correspond to the months when the temperature is exactly 75°F.

Step 5: Analyze the graph

Examine the graph to identify the points where it crosses the x-axis. These points represent when the temperature is exactly 75°F. Determine the intervals around these points where the graph is above the x-axis to find where the temperature exceeds 75°F.

Step 6: Interpret the graph results

Based on the graph, note the months (x-values) where the function \( g(x) \) is above the x-axis. These months are when the temperature is 75°F or more.

Key Concepts

Graphing Polynomials

Graphing polynomial functions is an intuitive way to visualize the changes in a function's output based on its input values. In this case, we have a polynomial function modeling temperatures. To graph this polynomial, you need to understand the shape and behavior of polynomials. Polynomials can have multiple curves, with up to "n-1" turning points where "n" is the degree of the polynomial. Our function is of the fourth degree, meaning it can have up to three peaks or valleys.

To graph:

• Start by identifying the degree and leading coefficient, as they influence the end behavior.
• Find the x-values (roots) where the polynomial may intersect the x-axis, which are the solutions to the equation.
• Plot enough points around the roots to see the full curve.

By observing these characteristics on a graph, we can make predictions and insights about when specific conditions are met, such as reaching or exceeding a temperature threshold.

Temperature Modeling

Temperature modeling using polynomials can provide valuable insights into seasonal trends and temperature fluctuations. The function we are examining estimates average high temperatures over a year at Daytona Beach, Florida. This approach allows us to easily visualize expected temperature patterns over different months.

• Advantages: Polynomial models can closely fit a wide range of data owing to their flexible nature.
• Flexibility: Allows for modeling complex seasonal trends and peaks.
• Visualization: Seeing the entire year's temperature change in a single graph can provide quick insights into peak temperature periods.

Temperature modeling helps predict future trends or identify significant temperature shifts. Analyzing where the graph rises above a certain temperature level can have real-world applications, such as planning for weather-dependent activities.

Intersection Points

Finding intersection points is a key concept when you set a polynomial equal to a specific value, like 75°F in this temperature modeling exercise. Intersection points represent the x-values where two equations meet. In graphing terms, for our function, these are the x-values where the graph of our temperature polynomial crosses the horizontal line at 75°F.

• Approach: Solve the equation by moving all terms to one side (resulting in a new polynomial set equal to zero).
• Graphical Method: Plot this new equation and identify where it intersects the x-axis.
• Interpretation: These intersection points indicate the months when the average temperature reaches exactly 75°F.

Analyzing the behavior around these intersection points helps you determine the range of months during which the temperature stays at or exceeds 75°F. This step is crucial in interpreting and utilizing the graph effectively.