Problem 89 - Algebra A Combined Function

Question

The average total daily supply \(y\) of motor gasoline (in thousands of barrels per day) in the United States for the period \(2000-2008\) can be approximated by the equation \(y=-10 x^{2}+193 x+8464,\) where \(x\) is the number of years after 2000. (Source: Based on data from the Energy Information Administration) a. Find the average total daily supply of motor gasoline in 2004 . b. According to this model, in what year, from 2000 to 2008 , was the average total daily supply of gasoline 9325 thousand barrels per day? c. According to this model, in what year, from 2009 on, will the average total supply of gasoline be 9325 thousand barrels per day?

Step-by-Step Solution

Answer

a) 9076, b) 2007, c) 2013.

Step 1: Substitute x for the Year

For part (a), 2004 is 4 years after 2000. Hence, substitute \(x=4\) into the equation.

Step 2: Calculate Value of y for 2004

Substitute \(x=4\) into the equation: \[ y = -10(4)^2 + 193(4) + 8464 \] Simplify to find \(y\). \[ y = -160 + 772 + 8464 = 9076 \] So, the supply of motor gasoline in 2004 is 9076 thousand barrels per day.

Step 3: Setup Equation for Given y

For part (b) and part (c), set \(y = 9325\) and solve for \(x\) using the equation: \[ -10x^2 + 193x + 8464 = 9325 \].

Step 4: Rearrange the Equation

Move all terms to one side of the equation to form a quadratic equation: \[ -10x^2 + 193x + 8464 - 9325 = 0 \], which simplifies to \[ -10x^2 + 193x - 861 = 0 \].

Step 5: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -10\), \(b = 193\), and \(c = -861\): \[ x = \frac{-193 \pm \sqrt{193^2 - 4(-10)(-861)}}{2(-10)} \].

Step 6: Calculate the Discriminant and x Values

Calculate the discriminant: \[ 193^2 - 4(-10)(-861) = 37249 - 34440 = 2809 \].Then find \(x\):\[ x = \frac{-193 \pm \sqrt{2809}}{-20} \]. \(\sqrt{2809} = 53\), hence:\[ x = \frac{-193 + 53}{-20} \] and \[ x = \frac{-193 - 53}{-20} \].

Step 7: Solve for x

Compute the two possible solutions for \(x\): 1. \[ x = \frac{-140}{-20} = 7 \] 2. \[ x = \frac{-246}{-20} = 12.3 \] (approximately)Thus, \(x = 7\) corresponds to 2007 and \(x = 12.3\) relates to 2012 when rounded to the nearest year, 12.3 indiciates the model's prediction of 2013.

Key Concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \). Solving these equations involves finding the value(s) of \( x \) that make the equation true. In the context of our exercise, we use the quadratic equation to model the supply of motor gasoline over time.
There are several methods to solve quadratic equations, including factorization, completing the square, and using the quadratic formula. In our scenario, the quadratic formula is particularly useful due to the nature and complexity of the equation involved.
When tackling a quadratic equation, it's important to first understand the equation and then plan the approach. Consider the specific methods that suit the equation, like using the quadratic formula when the equation does not easily factorize. By rearranging the terms to set the equation to zero, you make it ready for the most appropriate solving method. Keeping these steps organized ensures a smooth solution process.

Quadratic Formula

The quadratic formula is a robust tool for solving quadratic equations and is defined as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a \), \( b \), and \( c \) are coefficients in the equation \( ax^2 + bx + c = 0 \). This formula can find real or complex solutions for any standard quadratic equation.
In our gasoline supply exercise, the quadratic formula helps determine which years the average supply reaches a certain level, like 9325 thousand barrels. This involves calculating the discriminant, \( b^2 - 4ac \), a crucial part of the formula. The discriminant determines the nature of the solutions:

• If it's positive, there are two distinct real solutions.
• If zero, exactly one real solution exists.
• If negative, the equation has two complex solutions.

Calculating these values precisely allows us to predict years with specific gasoline supply levels, as demonstrated when applying the formula to solve for \( x \). This involves substituting the given numbers into the quadratic formula, simplifying to find \( x \), which corresponds to specific years.

Application of Quadratic Equations

Quadratic equations have a wide range of applications in different fields such as physics, finance, engineering, and as seen in this exercise, modeling real-world data, like gasoline supply over a period.
In this context, the quadratic equation models the trend of gasoline consumption, which helps predict future demand and assess past data.

• Part (a) allows us to calculate for a specific year, providing immediate insights through direct substitution.
• Part (b) and (c) utilize the model to determine when the supply meets specific conditions, indicating its predictive power.

This application demonstrates how mathematics extends beyond the classroom, illustrating essential concepts in understanding patterns and trends. When interpreting these models, it's critical to understand the assumptions and limitations our mathematical model might have, ensuring accurate application in practical scenarios. Thus, modeling with quadratic equations provides a powerful framework for analysis and decision-making based on historical data.