Problem 76 - Contemporary Precalculus

Question

The gross federal debt \(y\) (in trillions of dollars) in year \(x\) is approximated by $$ y=.79 x+3.93 \quad(x \geq 3) $$ where \(x\) is the number of years after \(2000 . \)Find the year in which the approximate federal debt is: \(\$ 14.2\) billion

Step-by-Step Solution

Answer

Answer: 1995

Step 1: Convert the given debt amount to trillions

Since y is in trillions of dollars, we need to convert \(14.2\) billion into trillions. To achieve this, divide the given amount by a million. So, $$ y = \frac{14.2\, billion}{1,000,000} = 0.0142\, trillion $$

Step 2: Substitute the value of y in the given equation

Now, substitute the value of y we obtained in Step 1 (\(0.0142\) trillion) into the given equation: $$ 0.0142 = 0.79x + 3.93 $$

Step 3: Solve for x

To find the value of x, we will first isolate x by subtracting 3.93 from both sides of the equation: $$ 0.0142 - 3.93 = 0.79x $$ After performing the subtraction, we get: $$ -3.9158 = 0.79x $$ Next, we will divide both sides by 0.79 to solve for x: $$ x = \frac{-3.9158}{0.79} $$ Upon calculating this division, we have: $$ x \approx -4.955 $$ It should be noted that although we have a negative value for x, we should keep in mind that x represents the number of years after 2000. Thus, a negative value indicates that the debt amount was reached before the year 2000.

Step 4: Calculate the year

Now that we have x, we can calculate the actual year by adding the resulting x-value to 2000: $$ Year = 2000 - 4.955 $$ After rounding the value of x to its nearest integer, we get: $$ Year = 2000 - 5 $$ Thus, the year in which the approximate gross federal debt reached \(14.2\) billion is: $$ Year = 1995 $$

Key Concepts

Linear Equations

Linear equations are a fundamental part of mathematics. They describe relationships where one quantity is a constant multiple of another, plus or minus a constant. In this context, the linear equation is used to model how federal debt changes over the years. The equation given is \( y = 0.79x + 3.93 \). This is a standard linear equation format, where:

• \( y \) represents the total debt in trillions,
• \( x \) represents the number of years after 2000,
• \( 0.79x \) is the slope, showing the annual increase in debt,
• \( 3.93 \) is the y-intercept, representing the debt in 2000.

The core idea here is understanding how to manipulate these types of equations to find unknown variables. Linear equations are a tool to predict future outcomes or understand past trends by plugging in known values and solving for the unknown.

Financial Mathematics

Financial mathematics is the application of mathematical methods to solve financial problems. In this problem, it's used to determine when a specific debt level was reached using linear equations. It involves working with large numbers and requires skills in:

• Converting financial amounts into the correct units for analysis,
• Interpreting mathematical models to understand financial changes over time,
• Solving equations to find specific financial outcomes such as past or future debt levels.

Financial mathematics is crucial for planning and forecasting. In this exercise, we predicted when a debt level of \( 14.2 \) billion dollars was achieved by solving for \( x \), which represents the years after 2000. The outcome helps in understanding financial trends and is useful for economists and policymakers.

Unit Conversion

Unit conversion is a critical process in mathematical modeling, especially in financial mathematics, where numbers often need to be expressed in more manageable terms. In this exercise, we needed to convert billions to trillions:

• We started by converting \( 14.2 \) billion dollars into trillions, because the equation uses trillions as the unit for debt measurement.
• To convert billions to trillions, divide the number by one million, as there are one thousand billion in a trillion.
• This conversion provides consistency with the units used in the linear equation, allowing accurate substitution and calculation.

Understanding how to convert between different units is a vital skill, ensuring that mathematical models are applied correctly, especially when dealing with very large or very small financial figures. Proper unit conversion allows for clear and precise financial analysis.