Problem 82 - College Algebra with Modeling and Visualization

Question

Continuous Compounding A sum of money \(P\) in an account receives continuous interest and triples in 15 years. Find a formula \(A(x)\) that calculates the amount of money in the account after \(x\) years.

Step-by-Step Solution

Answer

The formula is \( A(x) = Pe^{(\frac{\ln(3)}{15})x} \).

Step 1: Understanding Continuous Compounding

Continuous compounding refers to the process where the interest on an investment is calculated and added continuously, at every instant of time. The formula for continuous compounding is given by: \[ A = Pe^{rt} \] where \( A \) is the amount of money after time \( t \), \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time in years.

Step 2: Analyzing Given Conditions

We are given that the initial amount \( P \) triples in 15 years. So, the amount after 15 years, \( A \), will be \( 3P \). This gives us the equation: \[ 3P = Pe^{15r} \] We can use this equation to find the interest rate \( r \).

Step 3: Solving for the Interest Rate

To find the interest rate \( r \), divide both sides of the equation by \( P \):\[ 3 = e^{15r} \]Take the natural logarithm (ln) on both sides to solve for \( r \):\[ \ln(3) = 15r \]Thus, the interest rate \( r \) is:\[ r = \frac{\ln(3)}{15} \]

Step 4: Writing the Formula for Any \(x\) Years

Now we use the continuous compounding formula for any time \( x \) years. Substitute the value of \( r \) into the original formula:\[ A(x) = Pe^{\left( \frac{\ln(3)}{15} \right)x} \] This is the formula that calculates the amount of money in the account after \( x \) years.

Key Concepts

Exponential Growth

Exponential growth is a powerful concept in mathematics and finance that describes a process where a quantity doubles rapidly. In continuous compounding, the principal amount grows continuously at an exponential rate. This means the amount in the account increases by a fixed percentage over a specific period, leading to faster growth compared to simple interest.
To understand exponential growth in the context of our exercise, imagine your initial investment as a snowball that continuously receives more snow. Instead of adding a specific amount of snow at intervals, the snowball grows naturally and constantly, getting bigger each moment.
Here's why exponential growth makes a huge impact:

• It accelerates the growth of the investment because the growth rate applies to an ever-increasing base amount.
• The more time passes, the larger the growth, leading to potentially significant increases in value over time.

In our function, the number of times the amount triples is linked to the phrase "grows exponentially," as seen in the equation used: \[ A(x) = Pe^{(rx)} \] This formula reflects how the money amount grows exponentially with time, depending on the interest rate found in the exercise.

Interest Rate Calculation

Calculating interest rates accurately is crucial in determining how much your investment will grow over time. In continuous compounding, this involves finding the rate at which the principal amount increases exponentially.
In the given exercise, to find the interest rate, we used the fact that the money triples in 15 years. This sets up the equation \( 3 = e^{15r} \), where we solve for the interest rate \( r \).
Let's revisit how this calculation unfolds:

• First, isolate the exponential expression by dividing both sides of the equation by the principal \( P \).
• Then, apply the natural logarithm (\( \ln \)) to both sides. \( \ln(3) = 15r \) allows us to convert the exponential equation to a linear one.
• Solve for \( r \) by dividing \( \ln(3) \) by 15, resulting in \( r = \frac{\ln(3)}{15} \).

This value of \( r \) is crucial because it affects the exponential growth of the investment over any given time \( x \) years, as calculated in our final formula: \[ A(x) = Pe^{\left( \frac{\ln(3)}{15} \right)x} \].
Understanding interest rate calculation in this context shows how small differences in \( r \) greatly affect how quickly your investment grows.

Natural Logarithm (ln)

The natural logarithm, denoted by \( \ln \), is a special mathematical function used to simplify equations involving exponential growth and continuous compounding. The natural logarithm relates how long it takes for a given amount to reach a certain size through continuous growth.
When you take the natural logarithm of both sides of an equation involving \( e \), it effectively "undoes" the exponential, making complex calculations more manageable. For example, \(\ln(e^x) = x\).
In the exercise, using \( \ln \) allows us to find the continuous compounding interest rate. By applying it to \( 3 = e^{15r} \), we transform it into the linear equation \( \ln(3) = 15r \).
Here's a bit more about \( \ln \):

• It is based on the natural base \( e \) (~2.71828), which is fundamental in calculating exponential growth.
• \( \ln \) helps convert problems with exponential functions into simpler algebraic forms.
• It's often used in finance for calculating compound interest, growth rates, and more complex financial models.

The use of \( \ln \) demonstrates how mathematics simplifies real-world finance problems, making continuous compounding easier to understand and apply.