Problem 67 - College Algebra and Calculus: An Applied Approach

Question

\(P\) dollars, invested at interest rate \(r\) compounded annually, increases to an amount \(A=P(1+r)^{3}\) in 3 years. For an investment of \(\$ 1000\) to increase to an amount greater than \(\$ 1500\) in 3 years, the interest rate must be greater than what percent?

Step-by-Step Solution

Answer

The interest rate must be greater than approximately 14.469% for an investment of $1000 to increase to more than $1500 in 3 years.

Step 1: Understand the Given Variables

The exercise provides the following variables: the investment amount P = $1000, the final amount A = $1500, and the time period of 3 years.

Step 2: Substitute Variables into the Equation

Substitute the given values into the formula to get the equation \(1500 = 1000(1+r)^{3}\).

Step 3: Simplify the Equation

Simplify this equation by dividing both sides by 1000 which gives us \(1.5 = (1+r)^{3}\).

Step 4: Solve for interest rate (r)

To solve for \(r\), perform the cubic root on both sides, then subtract 1 from both sides we get \(r= \(\sqrt[3]{1.5}\) - 1.\) After solving this equation, we find that \(r \approx 0.14469\).

Step 5: Convert to percentage

Convert the decimal to a percentage to provide a more comprehensible answer, we get \(0.14469 \times 100 = 14.469%\). So, approximately a 14.469% annual compounded interest rate is needed for the investment to increase to more than $1500 in 3 years.

Key Concepts

Investment Growth

Investment growth refers to the increase in the value of an initial amount of money over a period of time. This growth occurs due to the interest earned or the appreciation of the asset's value. When we talk about investment growth in relation to compound interest, we're focusing on how invested money can increase at an accelerating rate. This happens because the interest is calculated not just on the principal amount, but also on any interest previously earned.

Compound interest is a key factor in investment growth, as it allows the investment to grow exponentially over time. Therefore, understanding how compound interest works is crucial to maximizing the potential returns on an investment.

For example, investing in a stable financial instrument with a high interest rate can lead to significant growth in your investment. However, it's important to remember that the duration of your investment also plays a critical role. The longer the investment period, the greater the potential for growth, assuming a constant rate of return.

Annual Interest Rate

The annual interest rate is the percentage at which your initial investment grows each year. It determines the amount of interest paid by the investment annually. This rate is typically provided as a percentage, and understanding it is essential for calculating investment growth.

In our example, we derived that an annual interest rate of approximately 14.469% is required to grow a $1000 investment to more than $1500 in three years using compound interest. This interest rate signifies how much more the investment will be worth after each year.
Here are some key points about the annual interest rate:

• A higher annual interest rate results in faster growth of your investment.
• The rate can vary depending on the type of investment and market conditions.
• It's essential to evaluate if the rate offered by an investment opportunity is suitable for your financial goals.

Calculating the right interest rate involves understanding how it compounds annually, affecting the ultimate growth of the investment. The compound interest equation reflects this by incorporating the rate into an exponential function.

Exponential Equations

Exponential equations are used to model situations where a quantity grows or decays at a rate proportional to its current value. In the context of compound interest, these equations are invaluable for predicting future investment values.

An exponential equation takes the form of \(A = P(1+r)^n\), where:

• \(A\) is the final amount after time \(n\).
• \(P\) is the principal amount or initial investment.
• \(r\) is the annual interest rate expressed as a decimal.
• \(n\) is the number of years the money is invested or borrowed.

The exponential equation reveals how every year, the amount of interest earned is added to the principal for the interest to be calculated on the new total the following year. This influences the final amount significantly, as seen from the exponential term \((1+r)^n\), highlighting the power of compound interest as opposed to simple interest.

Solving the equation can sometimes require specific mathematical operations, such as taking a root to isolate variables like the interest rate, as we did in the provided exercise. This highlights the importance of understanding exponential functions in financial calculations.