Problem 68 - Contemporary Precalculus

Question

The total resources (in billions of dollars) of the Pension Benefit Guaranty Corporation, the government agency that insures pensions, is approximated by $$ y=-.279 x^{2}+4.006 x+28.412 \quad(4 \leq x \leq 20) $$ where \(x=4\) corresponds to 2004 (a) When are resources the greatest? (b) Use the trace feature to find the approximate time when the Corporation will run out of money.

Step-by-Step Solution

Answer

Answer: The resources are the greatest around 2011, and the corporation will approximately run out of money around 2009.

Step 1: (Step 1: Identify the coefficients)

The quadratic function is given as: $$y = -0.279x^2 + 4.006x + 28.412$$ Let us identify the coefficients: a = -0.279 b = 4.006 c = 28.412

Step 2: (Step 2: Find the vertex)

We will use the formula: $$x = \frac{-b}{2a}$$ Plugging in the values of a and b: $$x = \frac{-4.006}{2(-0.279)}$$ Calculating the value of x: $$x \approx 7.19$$ Now substitute the x value back into the function to find the maximum resources: $$y = -0.279(7.19)^2 + 4.006(7.19) + 28.412$$ $$y \approx 42.55$$ Based on these calculations, the maximum resources ($42.55 billion) will be in the year: 2004 + 7.19 = 2011.19 or roughly 2011. (a) The resources are the greatest around 2011.

Step 3: (Step 3: Find the x-intercepts)

To find the x-intercepts, we need to set y = 0: $$0 = -0.279x^2 + 4.006x + 28.412$$ We can use the quadratic formula to solve for x: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Plugging in the values of a, b, and c: $$x = \frac{-4.006 \pm \sqrt{(4.006)^2 - 4(-0.279)(28.412)}}{2(-0.279)}$$ Calculating the two possible values for x: $$x \approx 4.71 \quad \text{or} \quad x \approx 20.46$$ However, since the given interval is \((4 \leq x \leq 20)\), the second value is not valid within this interval. So, we will only consider the first value: 2004 + 4.71 = 2008.71 or roughly 2009. (b) The corporation will approximately run out of money around 2009.

Key Concepts

Vertex of Parabola

In the context of quadratic functions, the **vertex** of a parabola is a pivotal point that can represent either the maximum or minimum value of the function, depending on its orientation. For a downward-opening parabola, which is the case in our problem (\(a = -0.279\), thus negative), the vertex represents the maximum point.
To find the vertex, you can use the formula: \[ x = \frac{-b}{2a} \]Here, \(b = 4.006\) and \(a = -0.279\). Plugging these into the formula gives us:\[ x \approx 7.19 \]This value indicates the time (in years from 2004) when the resources are at maximum. By substituting \(x = 7.19\) back into the function, we can find the exact maximum resources, which in our case is approximately $42.55 billion in the year 2011.
Thus, the vertex gives us essential information about the parabola's peak point in this scenario.

Quadratic Formula

The **Quadratic Formula** is crucial for finding the roots or \(x\)-intercepts of a quadratic equation like \(ax^2 + bx + c = 0\). The formula is represented by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula helps to determine where the function crosses the x-axis (if at all). In our provided solution, the quadratic equation used this formula to determine when the corporation resources will deplete.
By substituting \(a = -0.279\), \(b = 4.006\), and \(c = 28.412\) into the formula, two potential solutions for \(x\) are obtained:

• Approximately 4.71
• Approximately 20.46

However, within the practical interval \( 4 \leq x \leq 20 \), only the first solution is applicable, indicating that the resources will run out around the year 2009.
This demonstrates the versatility of the quadratic formula in predicting critical points for problem-solving.

Maximum and Minimum Values of Quadratics

The concepts of maximum and minimum values in quadratic equations are straightforward once you understand the parabola's orientation. A quadratic function opens upwards when \(a > 0\) and opens downwards when \(a < 0\).
For downward-opening parabolas, like our given problem, the vertex gives the maximum value. We've previously found that year 2011, denoted approximately by the vertex \(x \approx 7.19\), yields the maximum resources of roughly $42.55 billion.

• Maximum Value: Occurs at the vertex for downward parabolas.
• Minimum Value: Occurs at the vertex for upward parabolas.

Understanding when and where these extreme values occur aids immensely in interpreting quadratic behavior and ensuring accurate predictions in real-world applications.

x-intercepts

The **x-intercepts** of a quadratic function, also known as roots or zeros, are where the parabola cuts across the x-axis. These points are crucial in identifying when a particular scenario (like resources being zero) will occur.
Using the quadratic formula, \(x \approx 4.71\) was determined as one of the x-intercepts within the specified interval \(4 \leq x \leq 20\).
This indicates the year around 2009, when the corporation's resources are predicted to run out.
The method clearly shows the relevance of x-intercepts in problem-solving, where understanding where and when a function equals zero provides actionable insights.

Problem Solving with Quadratics

When it comes to **problem solving with quadratics**, understanding the properties and applications of quadratic equations is vital. In practical situations like predicting economic variables or scientific measurements, such equations are incredibly versatile.
This particular exercise involved using a quadratic function to determine both the maximum resources and the point where resources run out.

• First, identifying the vertex allowed us to find when resources peak at roughly $42.55 billion in 2011.
• Then, employing the quadratic formula identified the x-intercept where resources hit zero, approximately in 2009.

Both approaches combined give a comprehensive view of the problem, demonstrating the effectiveness of quadratic equations in real-life scenarios.
In general, solving problems with quadratics involves:

• Determining critical points like vertex or x-intercepts
• Using those points to make informed decisions or predictions

This systematic approach is what makes them so powerful and widely used in diverse areas.