Problem 71 - Precalculus : Building Concepts and Connections

Question

Find the interest rate \(r\) if the interest on the initial deposit is compounded continuously and no withdrawals or further deposits are made. Initial amount: $$ 8500 ;\( Amount in 5 years: $$ 10,000\)

Step-by-Step Solution

Answer

The interest rate, when calculated, is approximately equal to 3.31% when expressed as a percentage.

Step 1: Insert Knowns into Formula

First let's insert the given information into the formula \(A = P e^{rt}\). So we have, \(10,000 = 8500 \times e^{5r}\)

Step 2: Solving for \(r\)

To solve for \(r\) we first divide both sides of the equation by 8500 to isolate \(e^{5r}\) such that we get: \(\frac{10,000}{8500} = e^{5r}\). Then, take the natural log of both sides to remove the exponential term which will give: \(ln(\frac{10,000}{8500}) = 5r\)

Step 3: Calculate the Final Answer

Finally, to find the value of \(r\) we divide both sides by 5 leading to \(r = \frac{ln(\frac{10,000}{8500})}{5}\) and multiply this by 100 to convert the decimal figure into a percentage. Calculating this gives us the required interest rate.

Key Concepts

Natural Logarithm

A natural logarithm is a type of logarithm with a special base, specifically the number Euler's number (denoted as \( e \)), which is approximately equal to 2.71828. The natural logarithm is widely used in continuous compounding and exponential growth due to its properties that simplify many complex calculations. When you see \( \ln(x) \), it specifically means the power to which \( e \) has to be raised to obtain the number \( x \).

The process of using a natural logarithm helps us solve equations involving exponential functions. In our exercise, \( \ln \) is employed to "undo" the exponential function, making it possible to isolate the variable \( r \) (interest rate). Without using natural logarithms, solving exponential equations such as those found in continuous compounding would be much more difficult.

• Natural logarithms have a lot of applications, such as in growth calculations and decay trends.
• The natural logarithm of 1 is always 0: \( \ln(1) = 0 \).
• \( \ln(e) = 1 \) because \( e^1 = e \).

Exponential Growth

Exponential growth occurs when a quantity increases by a consistent percentage over equal intervals of time. It's a common model in fields like biology, finance, and technology. In the context of continuous compounding, it represents how investments grow over time when reinvestment happens continuously. This compounding leads to growth that accelerates as the quantity grows.

In our problem, the function \( A = Pe^{rt} \) shows exponential growth. Here:

• \( A \) is the amount in the future (final investment value).
• \( P \) is the initial principal (initial investment amount).
• \( r \) is the rate of growth (interest rate).
• \( t \) is time in years.

Exponential growth is powerful because even small interest rates can lead to significant increases over long periods. Unlike simple interest, which grows linearly, exponential growth compounds upon itself, leading to a steeper and steeper increase as time passes.

Interest Rate Calculation

Calculating the interest rate in continuous compounding involves using the exponential growth formula \( A = Pe^{rt} \). Here, determining the interest rate \( r \) can be a bit tricky because it involves reversing the effects of growth over time.

To find \( r \) for continuous compounding:

• First, divide the final amount \( A \) by the initial amount \( P \) to adjust the equation.
• Use a natural logarithm to "unpack" the exponent, which simplifies the exponential equation.
• Finally, divide by the time period \( t \) to isolate \( r \).

In our example, after rearranging the formula, \[ r = \frac{\ln(\frac{A}{P})}{t} \]It’s important to understand this step-by-step process because it allows you to determine the interest rate in situations where compounding is continuous. This tool is invaluable in financial decision-making, enabling you to compare growth across different scenarios.