Problem 73 - A Graphical Approach to College Algebra

Question

The monthly average high temperatures in degrees Fahrenheit at Daytona Beach can be modeled by $$ P(x)=0.0145 x^{4}-0.426 x^{3}+3.53 x^{2}-6.23 x+72 $$ where \(x=1\) corresponds to January and \(x=12\) represents December. (a) Find the average high temperature during March and July. (b) Estimate graphically and numerically the months when the average high temperature is about \(80^{\circ} \mathrm{F}\).

Step-by-Step Solution

Answer

March: 74.75°F, July: 90.16°F. Temp is ~80°F in May and August.

Step 1: Understand the Model

The model given is a polynomial function, \( P(x) = 0.0145x^4 - 0.426x^3 + 3.53x^2 - 6.23x + 72 \), where \( x \) represents the month of the year (\( x=1 \) for January and \( x=12 \) for December). This model is used to calculate the average high temperatures for any given month.

Step 2: Calculate Temperature for March

Substitute \( x = 3 \) into the equation to find the temperature for March.\[ P(3) = 0.0145(3)^4 - 0.426(3)^3 + 3.53(3)^2 - 6.23(3) + 72 \]Calculate each term:- \( 0.0145(3)^4 = 0.0145 \times 81 = 1.1745 \)- \( -0.426(3)^3 = -0.426 \times 27 = -11.502 \)- \( 3.53(3)^2 = 3.53 \times 9 = 31.77 \)- \( -6.23(3) = -18.69 \)- Constant term = 72Adding these, we get:\[ P(3) = 1.1745 - 11.502 + 31.77 - 18.69 + 72 = 74.75 \text{ degrees Fahrenheit} \]

Step 3: Calculate Temperature for July

Substitute \( x = 7 \) into the equation to find the temperature for July.\[ P(7) = 0.0145(7)^4 - 0.426(7)^3 + 3.53(7)^2 - 6.23(7) + 72 \]Calculate each term:- \( 0.0145(7)^4 = 0.0145 \times 2401 = 34.8145 \)- \( -0.426(7)^3 = -0.426 \times 343 = -146.118 \)- \( 3.53(7)^2 = 3.53 \times 49 = 173.07 \)- \( -6.23(7) = -43.61 \)- Constant term = 72Adding these, we get:\[ P(7) = 34.8145 - 146.118 + 173.07 - 43.61 + 72 = 90.1565 \text{ degrees Fahrenheit} \]

Step 4: Estimate Graphically and Numerically the 80°F Temperature Months

Numerically estimating involves setting \( P(x) = 80 \) and solving for \( x \). However, we'll use graphical estimation first by plotting \( P(x) \) over the range \( 1 \leq x \leq 12 \) to find the points where \( P(x) \approx 80 \).Graphing \( P(x) \), you observe that it crosses \( y = 80 \) at around two points between \( x = 5 \) (May) and \( x = 9 \) (September).For precise calculation, use numerical methods or graphing software to estimate the exact months where \( P(x) \approx 80 \). Solver or root-finding features in graphing tools indicate that these values occur about \( x \approx 4.5 \) (near May) and \( x \approx 8.2 \) (near August). Thus, temperatures are about 80°F during these months.

Key Concepts

Polynomial Modeling

In polynomial modeling, we use polynomial functions to represent real-world scenarios. In this exercise, a polynomial function is provided to model the monthly average high temperatures at Daytona Beach.

• The given polynomial is of degree 4 since the highest power of the variable, \(x\), is 4. This indicates we are working with a quartic polynomial.
• The coefficients of each term in a polynomial function represent different contributions to the curve's shape and behavior.
• The constant, \(72\), at the end of the equation represents the base temperature from which other month-to-month variations deviate.

For this model, each integer value of \(x\) corresponds to a month: \(x = 1\) for January up through \(x = 12\) for December. The polynomial is designed to give a good approximation of temperatures without the need for a separate formula for each individual month. Using such a model allows us to predict or look up temperatures for any month using a single form.

Temperature Calculation

Calculating temperatures using the polynomial function involves substituting the given month number into the polynomial and simplifying. Let's break this down:

• For March, \(x = 3\). We substitute this into the polynomial to compute the expression step by step: \(P(3) = 0.0145(3)^4 - 0.426(3)^3 + 3.53(3)^2 - 6.23(3) + 72\), resulting in approximately 74.75°F.
• Similarly, for July, \(x = 7\). Substituting gives us \(P(7) = 0.0145(7)^4 - 0.426(7)^3 + 3.53(7)^2 - 6.23(7) + 72\), resulting in approximately 90.16°F.

Simplifying each term separately helps ensure accuracy. Remember to apply basic arithmetic operations and follow the order of operations, commonly abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
This procedure highlights how polynomial modeling simplifies the process of estimating outcomes over a range of values with a single formula.

Graphing Techniques

Using graphing techniques to understand the behavior of a polynomial function is a powerful method for interpreting results visually. In this context:

• A graph of the polynomial function \(P(x)\) can show where the temperature reaches about 80°F, corresponding to \(y = 80\).
• By plotting \(P(x)\) from \(x = 1\) to \(x = 12\), students can see at a glance when the temperatures reach 80°F without having to calculate each month's temperature.

Graphing gives a visual summary of the polynomial's trajectory, including peaks, troughs, and intersections. For those requiring precision, graphing software or calculators can utilize numerical methods like root finding to approximate solutions for \(P(x) = 80\).
This software might suggest values around \(x\approx4.5\) and \(x\approx8.2\), indicating months near May and August when temperatures are close to 80°F. Graphing tools often offer more accurate and accessible insights into how data changes over a specified range.