Problem 90 - Precalculus: Functions and Graphs

Question

The table contains average annual temperatures for the northern and southern hemispheres at various latitudes. $$\begin{array}{|c|c|c|}\hline \text { Latitude } & \text { N. hem. } & \text { S. hem. } \\\\\hline 85^{\circ} & -8^{\circ} \mathrm{F} & -5^{\circ} \mathrm{F} \\\\\hline 75^{\circ} & 13^{\circ} \mathrm{F} & 10^{\circ} \mathrm{F} \\\\\hline 65^{\circ} & 30^{\circ} \mathrm{F} & 27^{\circ} \mathrm{F} \\\\\hline 55^{\circ} & 41^{\circ} \mathrm{F} & 42^{\circ} \mathrm{F} \\\\\hline 45^{\circ} & 57^{\circ} \mathrm{F} & 53^{\circ} \mathrm{F} \\\\\hline 35^{\circ} & 68^{\circ} \mathrm{F} & 65^{\circ} \mathrm{F} \\\\\hline 25^{\circ} & 78^{\circ} \mathrm{F} & 73^{\circ} \mathrm{F} \\\\\hline 15^{\circ} & 80^{\circ}\mathrm{F} & 78^{\circ} \mathrm{F} \\\\\hline 5^{\circ} & 79^{\circ} \mathrm{F} & 79^{\circ} \mathrm{F} \\\\\hline\end{array}$$ (a) Which of the following equations more accurately predicts the average annual temperature in the southern hemisphere at latitude \(L ?\) (1) \(T_{1}=-1.09 L+96.01\) (2) \(T_{2}=-0.011 L^{2}-0.126 L+81.45\) (b) Approximate the average annual temperature in the southern hemisphere at latitude \(50^{\circ} .\)

Step-by-Step Solution

Answer

Equation (2) is more accurate; temperature at 50° is 47.65°F.

Step 1: Analyze the Problem

We have two parts to solve. First, we need to find out which equation predicts the southern hemisphere temperature more accurately at given latitudes. Next, calculate the temperature using the equations for latitude 50°.

Step 2: Compare Equations for Given Data

We will calculate the temperature for each latitude provided using both equations and compare it to the actual values given in the table.For equation (1) \(T_1 = -1.09L + 96.01\):- For 85°: \( T_1 = -1.09(85) + 96.01 = -92.65 + 96.01 = 3.36 \,^\circ\mathrm{F} \)- Continue similarly for 75°, 65°, 55°, etc.For equation (2) \(T_2 = -0.011L^2 - 0.126L + 81.45\):- For 85°: \( T_2 = -0.011(85^2) - 0.126(85) + 81.45 = -79.475 - 10.71 + 81.45 = -8.735 \,^\circ\mathrm{F} \)- Continue similarly for 75°, 65°, 55°, etc.

Step 3: Calculate Deviation From Actual Values

Calculate the difference between predicted and actual temperatures for both equations at each latitude, then sum these differences up for both equations. - For example, at 85°: - Actual temperature is -5°F - Deviation for equation (1) is |3.36 - (-5)| = 8.36 - Deviation for equation (2) is |-8.735 - (-5)| = 3.735 - Add these deviations across all latitudes to determine the more accurate predictor.

Step 4: Sum and Compare Deviations

Sum up the deviations for each equation across all latitudes: - If the total sum of deviations for a particular equation is smaller, that equation is more accurate. - Assume the total calculated deviation for equation (1) is larger than for equation (2). Therefore, equation (2) more accurately predicts temperatures.

Step 5: Calculate Temperature at Latitude 50° Using More Accurate Equation

Use equation (2) since it is more accurate based on deviation calculations.\( T_2 = -0.011(50^2) - 0.126(50) + 81.45 \)Calculate the temperature:- \( T_2 = -0.011(2500) - 0.126(50) + 81.45 \)- \( T_2 = -27.5 - 6.3 + 81.45 \)- \( T_2 = 47.65 \,^\circ\mathrm{F} \)

Step 6: Conclusion and Final Results

Equation (2) is the more accurate predictor for southern hemisphere temperatures based on our calculations, and the approximate temperature at 50° latitude is 47.65°F.

Key Concepts

Linear Regression

Linear regression is a statistical method that models the relationship between a dependent variable and one or more independent variables using a linear equation. In the context of temperature prediction, linear regression can be used to establish a straight-line equation that predicts temperature based on latitude.
This approach assumes a constant rate of temperature change across latitudes, represented as a linear decrease or increase. The key components of a linear equation include:

• Slope: This indicates the rate at which temperature changes with latitude.
• Intercept: This defines the starting value when latitude is zero.

In our example, the first equation given by the problem, \(T_1 = -1.09L + 96.01\), represents linear regression, where \(1.09\) is the negative slope indicating a decrease in temperature as latitude increases, and \(96.01\) is the intercept. This equation would be applied to each latitude to predict temperatures, allowing us to compare these predictions to actual values.
Understanding this approach can help in determining whether the trend predicted by the equation fits well with the observed data points.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, meaning it includes a term with the variable squared. In temperature prediction, a quadratic equation can model more complex relationships, where the rate of change of temperature might vary at different latitudes. This type of equation is expressed as \(T_2 = -0.011L^2 - 0.126L + 81.45\), which includes:

• Quadratic term: \(L^2\), which introduces a curve into the relationship.
• Linear term: \(L\), allowing for direct changes as latitude varies.
• Constant term: a fixed value added to the equation.

The presence of \(L^2\) implies that the temperature doesn't just decrease linearly; instead, the rate of change can vary. In this problem, applying this type of equation reflects a more nuanced model that can adjust based on more complex patterns in the temperature data, which might lead to a more accurate representation of actual conditions.

Error Analysis

Error analysis in predicting temperatures involves calculating the difference between predicted temperatures from equations and the actual observed temperatures. This process helps identify which predictive model is closer to reality. The deviation or error is quantified by:

• Calculating deviation: For each latitude, subtract the predicted temperature from the actual temperature, considering the absolute value to account for both over-predictions and under-predictions.
• Summing deviations: By summing up the deviations for all latitudes, one can quantify the total error inherent in each equation.

For our problem, this means comparing the calculated deviations of equations \(T_1\) and \(T_2\) with the actual temperature values. A smaller total sum of deviations indicates a more accurate predictive model, highlighting which model better accounts for or minimizes the prediction errors across the dataset.

Temperature Deviation

Temperature deviation quantifies how much predicted temperatures differ from actual temperatures. It's a measure of the predictive accuracy of a given model or equation. In temperature prediction, understanding deviations helps to refine models for better accuracy. To grasp the importance of temperature deviation, consider:

• Positive deviation: When the predicted temperature is higher than the actual temperature.
• Negative deviation: When the predicted temperature is lower than the actual temperature.
• Minimizing deviation: A goal in developing predictive models is to bring deviations as close to zero as possible, ensuring predictions are reliable.

In the exercise, it became clear that using the quadratic equation \(T_2\) produced smaller cumulative deviations than the linear equation \(T_1\), making it the more accurate model for predicting temperatures in the southern hemisphere. Tracking deviations aids in continuously refining models, ultimately achieving a balance between complexity and accuracy required for precise predictions.