Problem 90 - Contemporary Precalculus

Question

The total resources \(T\) (in billions of dollars) of the Pension Benefit Guaranty Corporation, the government agency that insures pensions, can be approximated by the equation \(T=-.26 x^{2}+3.62 x+30.18,\) where \(x\) is the number of years after \(2000 .^{\dagger}\) Determine when the total resources are at the given level. (a) \(\$ 42.5\) billion (b) \(\$ 30\) billion (c) When will the Corporation be out of money \((T=0) ?\)

Step-by-Step Solution

Answer

Answer: The total resources will be at approximately - $42.5 billion around 2009 (8.651 years after 2000), - $30 billion around 2000 (very close to 0 years after 2000), - $0 billion (Corporation out of money) around 2016 (15.272 years after 2000).

Step 1: Write down the given total resource equation

We are given the total resources \(T\) equation: $$T = -0.26x^2 + 3.62x + 30.18$$

Step 2: Find x when T = \(42.5 billion

We will substitute \)T = 42.5\( into the equation: $$42.5 = -0.26x^2 + 3.62x + 30.18$$ Now we need to solve for \)x$.

Step 3: Solve the equation for x

Rearrange the equation to form a quadratic equation: $$0 = -0.26x^2 + 3.62x + (30.18 - 42.5)$$ Simplify the equation: $$0 = -0.26x^2 + 3.62x - 12.32$$ Solve the quadratic equation by either factoring, completing the square, or using the quadratic formula. In this case, we will use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Plugging in the values, we get: $$x = \frac{-3.62 \pm \sqrt{(3.62)^2 - 4(-0.26)(-12.32)}}{2(-0.26)}$$ Solve and simplify: $$x \approx 8.651$$ So, the total resources will be at \(\$42.5\) billion approximately 8.651 years after 2000, which means around 2009.

Step 4: Find x when T = \(30 billion

Repeat the process for \)T = 30$: $$30 = -0.26x^2 + 3.62x + 30.18$$ Since \(30.18\) is very close to \(30\), the value of x will be very close to \(0\), which means the total resources are approximately \(\$30\) billion in 2000.

Step 5: Find x when T = \(0 billion (Corporation out of money)

Finally, we will find when the total resources are \)0$: $$0 = -0.26x^2 + 3.62x + 30.18$$ Solve this quadratic equation by factoring or using the quadratic formula. Using the quadratic formula again: $$x = \frac{-3.62 \pm \sqrt{(3.62)^2 - 4(-0.26)(30.18)}}{2(-0.26)}$$ Solve and simplify: $$x \approx 15.272$$ So, the Corporation will be out of money approximately 15.272 years after 2000, which means around 2016.

Key Concepts

Quadratic Formula

The quadratic formula is a cornerstone of solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). When you have a quadratic equation that is difficult to factor or you want to calculate the exact roots, the quadratic formula is your tool of choice. It's given as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients from your quadratic equation and \(\pm\) means that there will typically be two solutions (or roots) to the equation: one when you add the square root term, and another when you subtract it.

This formula is derived from the process of completing the square and provides a systematic solution to any quadratic equation. It’s important to understand that the term \(b^2 - 4ac\) is called the discriminant and it determines the nature of the roots. If it's positive, you’ll have two real and distinct solutions. If it's zero, you’ll have exactly one real solution (also called a repeated root), and if it's negative, you’ll have two complex solutions.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in precalculus. There are several methods to solve them: factoring, completing the square, graphing, and using the quadratic formula. Each method has its own application depending on the nature of the equation. For example, factoring is quick and efficient if the quadratic easily decomposes into two binomials. However, not all quadratics can be factored neatly, and that's when other methods come in.

Using the Quadratic Formula

As mentioned, the quadratic formula is a versatile method that can solve any quadratic equation, regardless of the coefficients. In our Pension Benefit Guaranty Corporation problem, the coefficients didn't allow for easy factoring, so the quadratic formula was used to find the years after 2000 when resources would reach certain levels. This approach systematically yielded the answers without guesswork.

Mathematical Approximation

Mathematical approximation is crucial when dealing with real-world data that might not be entirely precise or when dealing with calculations that result in irrational numbers. In such cases, having a decimal or fraction that approximates the value closely enough is sufficient for practical purposes. We talk about different levels of approximation, from rounding to a certain number of decimal places to using significant figures depending on the situation.

For the Pension Benefit Guaranty Corporation problem, we used approximation to determine how many years after 2000 the corporation would run out of money. This is a realistic scenario where a precise number of days is not necessary; knowing the approximate year is useful enough for planning or reporting purposes. Approximations keep the numbers manageable and the calculations reasonable, especially when conveying information to a broader audience that might not require exact figures.