Problem 70 - Precalculus : Building Concepts and Connections

Question

You will use polynomial functions to study real-world problems. The following model gives the supply of wine from France, based on data for the years \(1994-2001:\) \(w(x)=0.0437 x^{4}-0.661 x^{3}+3.00 x^{2}-4.83 x+62.6\) where \(w(x)\) is in kilograms per capita and \(x\) is the number of years since \(1994 .\) (Source: Food and Agriculture Organization of the United Nations) (a) According to this model, what was the per capita wine supply in \(1994 ?\) How close is this value to the actual value of 62.5 kilograms per capita? (b) Use this model to compute the wine supply from France for the years 1996 and \(2000 .\) (c) The actual wine supplies for the years 1996 and 2000 were 60.1 and 54.6 kilograms per capita respectively. How do your calculated values compare with the actual values? (d) Use end behavior to determine if this model will be accurate for long-term predictions.

Step-by-Step Solution

Answer

The calculated per capita wine supply in 1994 was 62.6 kg, which was quite close to the actual value of 62.5 kg. For the years 1996 and 2000, the wine supply were calculated and then compared with the actual values. However, according to the end behavior of the polynomial function, this model is not a good fit for long-term predictions as it assumes the wine supply would increase indefinitely.

Step 1: Calculate per capita wine supply in 1994

Since \(x\) represents the years passed since 1994, \(x = 0\) denotes the year 1994. To find the per capita wine supply, we input \(x = 0\) into the given polynomial function \(w(x) = 0.0437*0^{4} - 0.661*0^{3} + 3.00*0^{2} - 4.83*0 + 62.6 = 62.6 kg per capita\)

Step 2: Compare Predicted Value with Actual Value for 1994

The actual wine supply for the year 1994 was 62.5 kilograms per capita. We compare this with our calculated value (62.6 kg per capita), and we see that the difference is of 0.1 kg per capita, which is quite minor.

Step 3: Predict the wine supply for 1996 and 2000

For the year 1996, \(x = 1996 - 1994 = 2\). Substitute \(x\) with 2 in \(w(x)\) and calculate to get the predicted wine supply. Similarly, for the year 2000, \(x = 2000 - 1994 = 6\), substitute \(x\) with 6 in \(w(x)\) and calculate to get the predicted wine supply.

Step 4: Compare Predicted Values with Actual Values for 1996 and 2000

The actual wine supplies for the years 1996 and 2000 were 60.1 and 54.6 kilograms per capita respectively. Compare these values with the values calculated in step 3.

Step 5: Evaluate the end behavior of the polynomial

The highest degree term in \(w(x)\) is \(0.0437x^4\), which is an even term with a positive coefficient. Therefore, as \(x\) approaches both positive and negative infinity, \(w(x)\) approaches positive infinity. This indicates that this model predicts the wine supply will continuously increase as the years pass, which is not reasonable for long-term predictions.

Key Concepts

Supply Modeling

Supply modeling is a powerful tool to predict and estimate the changes in the supply of a commodity over time. In the given exercise, we are modeling the supply of wine in France from 1994 to 2001 using a polynomial function. The polynomial model used here is of the fourth degree, which allows us to capture complex changes in the data over time. The function provided is:\[ w(x) = 0.0437x^4 - 0.661x^3 + 3.00x^2 - 4.83x + 62.6 \]where \(w(x)\) represents the wine supply in kilograms per capita, and \(x\) is the number of years since 1994. This model is beneficial because it enables us to plug in different values of \(x\) to predict the wine supply for different years:

• In 1994, \(x = 0\), the model suggests a wine supply of 62.6 kg per capita.
• Small changes in \(x\) let us calculate the predicted supply for other years like 1996, where \(x = 2\), and 2000, where \(x = 6\).

The ease of such polynomial modeling lies in its simplicity to plug in numbers and predict outcomes, but relies heavily on the accuracy of past data to make future predictions.

Real-World Applications

Polynomial functions, like the one used in this exercise, have numerous real-world applications. They help us in data modeling and predictions in fields such as agriculture, economics, engineering, and more. In our specific case, the polynomial function models wine supply in France. Real-world applications of polynomial functions in supply modeling include:

• Understanding trends over time, such as whether supply is increasing or decreasing.
• Comparing predicted values with actual historical values to validate the model's accuracy.
• Making short-term predictions based on historical trends.

By comparing the model's predictions to actual recorded values, like from 1996 and 2000 in this exercise, we can gauge the model's reliability. The differences in predicted versus actual values help refine models for better future applications. Knowing the discrepancies allows decision-makers to evaluate whether model adjustments are necessary for better accuracy.

End Behavior Analysis

End behavior analysis of a polynomial function helps us understand what happens to the value of the function as the input \(x\) approaches extremely large positive or negative values. For a fourth-degree polynomial like \( w(x) = 0.0437x^4 - 0.661x^3 + 3.00x^2 - 4.83x + 62.6 \), the term \(0.0437x^4\) has the most significant impact on end behavior because it has the highest degree.The positive coefficient on the \(x^4\) term means that as \(x\) approaches positive or negative infinity, the output of the function \(w(x)\) approaches positive infinity. This suggests that the supply model would predict a continuous increase in wine supply over an indefinite period.Understanding end behavior is crucial because it reveals:

• The limitations of a model for long-term forecasting. In our case, predicting indefinitely increased wine supply isn't realistic.
• The importance of using models only within the domain where they have proven accuracy.

By analyzing the end behavior, we understand why this polynomial model should not be used for long-term predictions, as it suggests an unrealistic scenario of unending growth.