Problem 88 The monthly average high tempera... [FREE SOLUTION]

Chapter 4: Problem 88

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.

Short Answer

Expert verified

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

Step by step solution

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掳F, using a graphical approach.

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掳F. Set up the equation \( f(x) = 75 \). This gives us: \[ 0.0145x^4 - 0.426x^3 + 3.53x^2 - 6.22x + 72 = 75 \].

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掳F.

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掳F.

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掳F. Determine the intervals around these points where the graph is above the x-axis to find where the temperature exceeds 75掳F.

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掳F or more.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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掳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掳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掳F.

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

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Short Answer

Step by step solution

Understand the problem

Set up the equation for graphing

Convert equation to find intersection

Graph the function

Analyze the graph

Interpret the graph results

Key Concepts

Graphing Polynomials

Temperature Modeling

Intersection Points

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