Problem 58 - Algebra A Combined Function

Question

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In 2000 , crude oil field production in the United States was 2130 thousand barrels. In \(2007,\) U.S. crude oil field production dropped to 1850 thousand barrels. (Source: Energy Information Administration) a. Write two ordered pairs of the form (years after 2000, crude oil production). b. Assume the relationship between years after 2000 and crude oil production is linear over this period. Use the ordered pairs from part (a) to write an equation of the line relating years after 2000 to crude oil production. c. Use the linear equation from part (b) to estimate crude oil production in the United States in 2010 , if this trend were to continue.

Step-by-Step Solution

Answer

a. (0, 2130) and (7, 1850) b. Equation: \(y = -40x + 2130\) c. 1730 thousand barrels.

Step 1: Identify ordered pairs

First, convert the given years into the format \((x, y)\), where \(x\) is the number of years after 2000, and \(y\) is the crude oil production in thousand barrels. For 2000, the year value is 0 (since 2000 - 2000 = 0), so the pair is (0, 2130). For 2007, the year value is 7 (since 2007 - 2000 = 7), resulting in the pair (7, 1850).

Step 2: Calculate the slope

Use the formula for the slope \(m\) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\): \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the ordered pairs \((0, 2130)\) and \((7, 1850)\) into the formula:\[m = \frac{1850 - 2130}{7 - 0} = \frac{-280}{7} = -40\]

Step 3: Write the equation in slope-intercept form

The slope-intercept form of a line is given by \(y = mx + b\). Using the point \((0, 2130)\) and the slope \(-40\) calculated in Step 2, substitute into the equation:\[2130 = -40(0) + b\]So, \(b = 2130\), making the equation:\[y = -40x + 2130\]

Step 4: Estimate crude oil production in 2010

To estimate the crude oil production for the year 2010, find \(x\) using the years since 2000: 2010 - 2000 = 10. Substitute \(x = 10\) into the linear equation:\[y = -40(10) + 2130 = -400 + 2130 = 1730\]Thus, the estimated production in 2010 is 1730 thousand barrels.

Key Concepts

Linear Relationship

When discussing a linear relationship, we're exploring how two quantities change in relation to each other. In mathematics, a linear relationship is one where the change in one variable consistently corresponds to a change in another. This means if you were to graph these two variables, you'd get a straight line.
The linear relationship can be represented algebraically by an equation of a line, often expressed in the slope-intercept form: \( y = mx + b \). Here, \( m \) represents the slope, which indicates the steepness of the line, and \( b \) denotes the y-intercept, where the line crosses the y-axis.
Understanding linear relationships is crucial because they make it simple to predict unknown values based on known data, like estimating future trends or values.

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics that denote specific points on a graph. Each ordered pair is comprised of two elements: \((x, y)\). The first element \(x\) represents the position on the horizontal axis, while \(y\) indicates the position on the vertical axis.
In the context of our exercise, when dealing with years and crude oil production, the ordered pairs such as \((0, 2130)\) for the year 2000 and \((7, 1850)\) for the year 2007 provide precise points that illustrate the relationship between time and production.
By using ordered pairs, we can plot these data points on a graph and determine the nature of their relationship, which in this case is linear.

Slope Calculation

Calculating the slope of a line is an essential step in understanding a linear relationship. The slope is a measure of how much the dependent variable \(y\) changes with a one-unit change in the independent variable \(x\). It's calculated as the ratio of the change in \(y\) to the change in \(x\):\[ m = \frac{y_2 - y_1}{x_2 - x_1}\]
In our exercise, we found the slope using the ordered pairs \((0, 2130)\) and \((7, 1850)\). We calculated:\[ m = \frac{1850 - 2130}{7 - 0} = -40\]
This negative slope suggests that as the years increase, the crude oil production decreases at a constant rate of 40 thousand barrels per year.

Linear Equation Estimation

Estimating values using a linear equation is a handy tool in prediction. Once we have a linear equation, like the one derived from our exercise \(y = -40x + 2130\), we can estimate unknown values.
Let's consider estimating crude oil production for a future year. Say we want to know the production in 2010, which is 10 years after 2000. We substitute \(x = 10\) into our equation: \[y = -40(10) + 2130 = 1730\]
This gives us an estimation of 1730 thousand barrels in 2010.
This methodology of using linear equations allows for easy and quick estimation in many real-world situations, making it a valuable tool for analysts and decision-makers.