Problem 77 - College Algebra with Modeling and Visualization

Question

Federal Debt In fiscal year 2008 the federal budget deficit was about \(\$ 340\) billion. At the same time, 30 -year treasury bonds were paying \(4.54 \%\) interest. Suppose the American taxpayer loaned \(\$ 340\) billion to the federal government at \(4.54 \%\) compounded annually. If the federal government waited 30 years to pay the entire amount back, including the interest, how much would this be?

Step-by-Step Solution

Answer

The federal government would owe approximately \(\$ 1272 \) billion after 30 years.

Step 1: Identify the Formula

To find the total amount owed after 30 years with interest compounded annually, we use the compound interest formula: \[ A = P(1 + r)^n \]where \( A \) is the amount in the future, \( P \) is the principal amount (initial amount), \( r \) is the annual interest rate, and \( n \) is the number of years.

Step 2: Substitute the Given Values

We have the following values:- Principal, \( P = 340 \) billion dollars,- Interest rate, \( r = 4.54\% = 0.0454 \), - Number of years, \( n = 30 \).Substitute these values into the compound interest formula: \[ A = 340(1 + 0.0454)^{30} \]

Step 3: Calculate the Compound Factor

Calculate the value of the compound factor, \((1 + 0.0454)^{30}\):\[ 1 + 0.0454 = 1.0454 \]Now compute: \[ 1.0454^{30} \approx 3.7409 \]

Step 4: Calculate the Future Amount

Substitute the compound factor back into the amount formula:\[ A = 340 \times 3.7409 \approx 1271.906 \]

Key Concepts

Future Value Calculation

Future value calculation is an essential concept in financial mathematics. It's the process of determining how much an investment made today will grow over a specified period. This involves using the principle of compound interest, where the interest earned each period is added to the principal and itself earns interest in subsequent periods. This is fundamentally different from simple interest, where interest is only earned on the initial principal.

The compound interest formula is: \[ A = P(1 + r)^n \] Here:

• \( A \) = Future Value (the amount of money you'll have in the future)
• \( P \) = Principal (the initial amount of money you invest)
• \( r \) = Annual interest rate (expressed as a decimal)
• \( n \) = Number of compounding periods (years)

In our exercise, this formula helps us calculate how much the federal government would owe after 30 years, considering the interest compounded annually. This type of calculation is useful not just for loans, but also for investments, savings plans, and other financial situations.

Exponential Growth

Exponential growth is a powerful concept that describes how quantities increase rapidly over time. This kind of growth occurs in situations where the growth rate is proportional to the current value, leading to faster increases as time progresses. In financial terms, this is seen with compound interest, where the growth of an investment accelerates as the interest continually increases the principal.

The expression \((1 + r)^n\) in the compound interest formula represents exponential growth. Here, the base \(1 + r \) shows how much the quantity grows per period, while the exponent \( n \) determines the number of periods the growth will occur. It's exciting because the power of earning interest on interest leads to significant growth over time, especially with long-term investments.

Financial Mathematics

Financial mathematics is a field that applies mathematical methods to solve problems in finance. It incorporates concepts like interest rates, annuities, loans, investments, and their associated calculations. Understanding these principles is vital for making informed decisions about savings, spending, investing, and borrowing.

This discipline enables individuals and businesses to analyze financial patterns and predict future trends. It also teaches how to maximize returns and minimize risks by properly managing financial resources. The compound interest calculation used in the exercise is a prime example of financial mathematics in action, showcasing how these mathematical techniques help predict future financial situations, be it for loans or investments.