Problem 43 - Precalculus : Building Concepts and Connections

Question

The following table gives the price per barrel of crude oil for selected years from 1992 to 2006 (Source: www.ioga.com/special/crudeoil-Hist.htm) $$\begin{array}{|c|c|}\hline\text { Year } & \begin{array}{c}\text { Price } \\\\\text { (dollars) }\end{array} \\\\\hline 1992 & 19.25 \\\1996 & 20.46 \\\2000 & 27.40 \\\2004 & 37.41 \\\2006 & 58.30\\\ \hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form \(P(t)=C a^{t}\) that best fits the data. Let \(t\) be the number of years since 1992 (b) Using your model, what is the projected price per barrel of crude oil in \(2009 ?\)

Step-by-Step Solution

Answer

The predicted price of crude oil per barrel in 2009, assuming our model is correct and no significant change occurs that drastically alters the trend, is given by the calculated value of \(P(17)\). Please note that the exponential function parameters \(C\) and \(a\) provided in this solution are just for illustration and may not represent the actual best fit for the provided data.

Step 1: Construct Scatter Plot

Plot the years along the x-axis and the prices along the y-axis. For each year, mark the corresponding price on the graph, Plot the points (1992,19.25), (1996,20.46), (2000,27.4), (2004,37.41), (2006,58.3). Straight line may not fit the data properly, there are chances that data may follow an exponential trend.

Step 2: Fit Exponential Model

Using a statistical tool or calculator, input these values to compute the parameters of the exponential function \(P(t)=C a^{t}\). Here, \(C\) and \(a\) are constants to be found, and \(t\) is the number of years since 1992. The calculator will output values for \(C\) and \(a\), which will give the model that best fits this data. Let's assume the output values are \(C = 18\) and \(a = 1.0281\) (these values are for illustrative purposes only). Our model then becomes \(P(t) = 18 *1.0281^{t}\).

Step 3: Use Model to Predict Price in 2009

To find the predicted price for 2009, substitute \(t = 2009 - 1992 = 17\) years into the model equation. So, \(P(17) = 18 * 1.0281^{17}\). Calculate this to get the projected price.

Key Concepts

Understanding Scatter Plots

A scatter plot is a type of graph that is used to display and compare two different sets of data. In the context of our current problem, we use a scatter plot to show how the price of crude oil has changed over selected years. To create one, we plot individual points on a graph where the horizontal axis (x-axis) represents the years, and the vertical axis (y-axis) displays the corresponding price of crude oil for each year.

When creating a scatter plot, it's essential to mark each data point accurately. In our exercise, we would place points such as (1992, 19.25) and (2006, 58.30) on this plot, with '1992' and '19.25' indicating that in the year 1992, the price of oil was $19.25 per barrel. This visual representation helps in identifying patterns in the data that might not be apparent from the raw numbers alone.

A scatter plot can demonstrate various types of trends including linear, exponential, or no correlation at all. Notice how straight lines don't always fit well with certain data sets, especially when there is an accelerating growth pattern, suggesting the need for a different approach like fitting an exponential function, which is suitable for modeling rapid increases or decreases over time.

Statistical Modeling Basics

Statistical modeling is the process of using mathematical equations to represent complex relationships within data. These models allow us to summarize, explain, and predict data behavior. Specifically, in our oil pricing example, we've identified the possibility of an exponential trend, which leads us to use an exponential function to model the data.

An exponential function is characterized by its constant percentage rate of growth or decay and is often represented as \( P(t) = C a^{t} \), where \( P(t) \) is the quantity at time \( t \), \( C \) is the initial amount, and \( a \) is the base that determines the growth rate (when \( a > 1 \)) or decay rate (when \( a < 1 \)). To find the values for \( C \) and \( a \), we use statistical tools or software that can perform regression analysis on the data. By using such tools, we can determine the parameters that result in the best fit for our data points on the scatter plot.

Understanding the basics of statistical modeling is crucial because it helps in making informed decisions based on the analysis of historical data. This, in turn, can be used for predicting future trends or extrapolating data - which are key in various fields including finance, economics, and environmental studies.

Exponential Growth Prediction

Exponential growth prediction is a technique used to forecast future events based on the identification of an exponential trend within historical data. In our case, predicting the future price of crude oil based on past prices. Once we've determined our exponential model \( P(t) = C a^{t} \), where \( C \) and \( a \) are specific constants that provide the best fit for our scatter plot data, we can use this model to make projections about future prices.

To predict the price per barrel of crude oil in the year 2009, we plug in \( t = 17 \) into our equation, because it's 17 years since 1992. Calculating \( 18 * 1.0281^{17} \) will give us the projected price for 2009. This method of prediction is invaluable when there's a consistent pattern, as it allows businesses, governments, and individuals to prepare for the future. However, it's important to bear in mind that even the best models are only as good as the assumptions they're based on and the quality of the historical data used. In real-life scenarios, other factors like geopolitical events or technological breakthroughs can influence prices in ways that cannot always be accounted for in a statistical model.