Problem 13 - Calculus and its applications

Question

Euler Bank advertises that it compounds interest continuously and that it will double your money in 15 yr. What is its annual interest rate?

Step-by-Step Solution

Answer

The annual interest rate is approximately 4.62%.

Step 1: Identify the formula for continuous compounding

The formula for continuously compounded interest is given by \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate, and \( t \) is the time in years.

Step 2: Set up the problem

We know the money will double, so \( A = 2P \) and \( t = 15 \) years. The equation becomes \( 2P = Pe^{15r} \). Simplify this to \( 2 = e^{15r} \) by dividing both sides by \( P \).

Step 3: Solve for \( r \)

To find \( r \), take the natural logarithm of both sides: \( \ln(2) = 15r \). Thus, \( r = \frac{\ln(2)}{15} \).

Step 4: Calculate the interest rate

Using a calculator, find \( \ln(2) \approx 0.693 \). Thus, \( r = \frac{0.693}{15} \approx 0.0462 \), which is 4.62%.

Key Concepts

Interest Rate Calculation

Calculating interest rates, especially in the realm of continuously compounded interest, is a fascinating topic that revolves around a specific formula: \( A = Pe^{rt} \). In this formula:

• \( A \) represents the accumulated amount after time \( t \)
• \( P \) is the principal or initial amount
• \( r \) stands for the annual interest rate
• \( t \) is the time in years

Setting up the problem:
When knowing that your investment doubles, you plug \( A = 2P \) into the equation. This situation implies you are solving for when \( 2P = Pe^{rt} \) under the condition \( t = 15 \) years.
After simplifying, dividing both sides by \( P \) gets you to \( 2 = e^{15r} \). Next, you'll solve for \( r \) using the properties of natural logarithms.

Natural Logarithm

Natural logarithms are a key mathematical concept used in solving equations related to exponential growth and decay, like those in continuously compounded interest.
Understanding the natural logarithm:
A logarithm is essentially the inverse of an exponent. A natural logarithm, symbolized as \( \ln \), specifically uses the base \( e \), which is an irrational constant approximately equal to 2.71828.
Applying natural logarithm in interest calculations:
To solve the equation \( 2 = e^{15r} \), take the natural logarithm on both sides to obtain \( \ln(2) = 15r \).
This simplifies our work: you use calculator tools to find that \( \ln(2) \approx 0.693 \). Therefore, you calculate the interest rate \( r \) using the expression \( r = \frac{\ln(2)}{15} \), leading us to an exact interest rate solution.

Doubling Time

Doubling time is a practical concept that indicates how long it will take for an investment to double in value at a certain continuous interest rate. Understanding this concept helps in both short-term and long-term financial planning.
How doubling time is derived in continuous compounding:
It's all about rearranging the formula \( A = Pe^{rt} \) with \( A = 2P \) when accounting for doubling, and it simplifies under the continuous compounding formula scenario.
You reach the equation \( 2 = e^{15r} \), and by employing the natural logarithm, you derive \( \ln(2) = 15r \). This steps into finding the rate \( r = \frac{\ln(2)}{t} \), where \( t \) is the doubling time, given as 15 years in this example.
Calculating the time:
To reverse this thought, you can also determine the doubling time if the interest rate is known, using \( t = \frac{\ln(2)}{r} \), showing this vital relationship between time, rate, and doubling.