Problem 17 - Algebra 2 and Trigonometry

Question

When interest is compounded quarterly (4 times a year) at an annual rate of 6\(\%\) , the rate of interest for each quarter is \(\frac{0.06}{4}\) , and the number of times that interest is added in \(t\) years is 4\(t\) . After how many years will an investment of \(\$ 100\) compounded quarterly at 6\(\%\) annully be worth at least \(\$ 450 ?\) (Use the formula \(A_{n}=A_{0}\left(1+\frac{r}{n}\right)^{n t} . )\)

Step-by-Step Solution

Answer

The investment will be worth at least $450 in 26 years.

Step 1: Understanding the Formula

We are using the compound interest formula \( A_n = A_0 \left(1 + \frac{r}{n}\right)^{nt} \), where \( A_n \) is the future value, \( A_0 \) is the initial investment, \( r \) is the annual interest rate, \( n \) is the number of times interest is compounded per year, and \( t \) is time in years.

Step 2: Identifying Given Values

From the problem, we have \( A_0 = 100 \), \( A_n = 450 \), \( r = 0.06 \), and \( n = 4 \). We need to find \( t \) such that the initial investment grows to at least \$450.

Step 3: Setting Up the Inequality

We want the future value \( A_n \) to be at least \$450. So, substitute into the formula: \[ 450 \leq 100 \times \left(1 + \frac{0.06}{4}\right)^{4t} \] Simplify inside the parentheses: \[ 450 \leq 100 \times \left(1.015\right)^{4t} \]

Step 4: Solving for 4t in Inequality

Divide both sides by 100 to isolate the exponential term:\[ 4.5 \leq 1.015^{4t} \]

Step 5: Applying Logarithms to Isolate t

Take natural logarithms on both sides to solve for \( t \): \[ \ln(4.5) \leq \ln(1.015^{4t}) \] This simplifies to: \[ \ln(4.5) \leq 4t \cdot \ln(1.015) \]

Step 6: Solving for t

Divide both sides by \( 4 \cdot \ln(1.015) \) to solve for \( t \):\[ t \geq \frac{\ln(4.5)}{4 \cdot \ln(1.015)} \] Calculate the values: \( \ln(4.5) \approx 1.504 \), and \( \ln(1.015) \approx 0.0149 \). Thus, \[ t \geq \frac{1.504}{0.0596} \approx 25.23 \]

Step 7: Conclusion

Since \( t \) must be an integer, round up to the nearest whole number. Hence, it takes at least 26 years for the investment to be worth at least \$450.

Key Concepts

Quarterly Compounding

Quarterly compounding is a method used in finance to calculate how often interest is added to the principal amount each year. When interest is compounded quarterly, it means that the interest is calculated and added to the principal four times a year. This is different from annual compounding, where interest is added just once. Here's how it works:

• Every quarter, or three months, the interest is added to the principal.
• This means each quarter, you effectively earn interest on interest, which leads to higher returns compared to annual compounding.

By breaking the year into four periods, each period sees the principal grow slightly from the previous one, leading to substantial growth over time. In our exercise, with an annual interest rate of 6%, each quarter earns 1.5%, calculated as 6% divided by 4, because the interest is spread over four periods.

Annual Interest Rate

An annual interest rate is the percentage increase on your investment or savings over the course of a year. It is essential in understanding how much return you can expect from an investment annually, but sometimes it’s not applied all at once.

• Many investments and loans use annual rates but apply them more frequently, such as quarterly, as in our exercise.
• To find the quarterly rate from the annual rate, you divide the annual rate by the number of compounding periods in a year.

In our exercise, this means we divide the 6% annual rate by 4, resulting in a 1.5% rate per quarter. The more frequently interest is compounded, the more total interest you earn over time, making the compounding frequency a crucial factor in your investment’s growth.

Exponential Growth

Exponential growth is a concept where quantities grow increasingly faster at a constant rate per period. This is often observed in finance with investments that compound over time. Let's consider how this applies:

• When you invest money, each time interest is added (compounded), it increases the base amount for future interest calculations.
• This results in exponential growth, meaning the bigger the principal becomes, the more significant each interest gain is.

In mathematical terms, exponential growth for compound interest is represented by the formula: \( A_n = A_0\left(1 + \frac{r}{n}\right)^{nt} \), where each component contributes to growing your principal exponentially over time. As more compounding periods take place, even a small interest rate results in rapid increases in total returns.

Natural Logarithms

Natural logarithms, denoted \(\ln\), are a mathematical function used to unravel exponential equations. In finance, it's particularly useful when trying to determine how long it will take for an investment to reach a particular value.

• Using natural logarithms helps us reverse equations where the variable (like time, \(t\)) is in the exponent, which is common in compound interest problems.
• In our exercise, the natural logarithm was used to solve for the time \(t\) when isolating it from the compound interest formula.

When you reach an equation like \( \ln(4.5) \leq 4t \cdot \ln(1.015) \), taking the natural log allows us to solve for \(t\), the time it will take for an investment to grow to a specified amount, which is the fundamental goal of using logarithms in these finance equations.