Problem 109 - Introductory and Intermediate Algebra for College Students 4th

Question

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 2350 & & \text { Triple the amount invested } & 7 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The annual interest rate needed to triple an investment of $2350 in 7 years with continuous compounding is approximately equal to \(ln(3) / 7\) or about 0.1429 in decimal form.

Step 1: Rephrase the Problem

The first step to solving this problem is to rephrase it in terms of the formula for continuous compounding. In this case, the accumulated amount is triple the initial investment, or $7050 (\$2350 * 3). So the equation we need to solve is \(7050 = 2350 * e^{7r}\).

Step 2: Isolate the Exponential Term

To isolate the variable 'r', we first need to isolate the exponential term. This means we need to divide both sides of the equation by $2350. Doing so gives us: \(e^{7r} = 3\).

Step 3: Apply the Natural Logarithm

Applying the natural logarithm 'ln()' to both sides of the equation allows us to remove the 'e' and gives us: \(ln(e^{7r}) = ln(3)\) which simplifies to \(7r = ln(3)\).

Step 4: Solve for r

Finally, we can solve for the variable 'r' by dividing both sides of the equation by 7, yielding \(r = ln(3) / 7\).

Key Concepts

Exponential Growth Explained

When we talk about exponential growth, we’re looking at how quantities can increase rapidly due to the compound effect of growth on top of growth. This is a concept that appears frequently in finance, population dynamics, and even in the spread of diseases. Imagine you’re watching a lily pad grow. At first, it doubles in size every day. By the tenth day, the growth is immense, despite starting out small.

Mathematically, exponential growth is described by the equation \( A = P e^{rt} \), where \( A \) is the amount after time \( t \) has passed, \( P \) is the principal amount (the initial quantity), \( r \) is the growth rate, and \( e \) is the base of the natural logarithm. The \( e^{rt} \) part is crucial—it’s what causes the 'snowball effect' of growth. As time increases, the \( e^{rt} \) term grows very large, very quickly, leading to a steep, upward curve when graphed.

Understanding the Natural Logarithm

In mathematics, the natural logarithm is a way to reverse the process of exponential growth, essentially 'unpacking' the exponent back into a multiple. The natural logarithm is denoted as \( ln(x) \) and is the inverse of the exponential function \( e^x \) when the base is \( e \) (approximately equal to 2.71828).

When you take the natural logarithm of an exponential expression like \( e^{7r} \) in the exercise, it simplifies to the power itself, which in this case is \( 7r \). This property \( ln(e^x) = x \) is incredibly useful since it allows us to solve for exponents, which is often otherwise difficult or impossible with algebra alone. The exercise demonstrates the use of this property nicely—the natural logarithm simplifies the equation and makes it possible to isolate and solve for the variable \( r \) representing the annual interest rate in the context of continuous compounding.

The Compound Interest Formula and Continuous Compounding

Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It's a fundamental concept in finance that's used to calculate the accrued interest over time. The generic compound interest formula is \( A = P(1 + \frac{r}{n})^{nt} \) where \( A \) is the amount of money accrued after \( n \) years, including interest, \( P \) is the principal amount, \( r \) is the annual interest rate (in decimal), and \( n \) is the number of times that interest is compounded per year.

However, when you have continuous compounding, the number of times the interest is compounded grows infinitely large. Mathematically, we express this as \( A = Pe^{rt} \), where \( e \) is again Euler's number. This exponential function models continuous growth and can lead to much greater amounts over time than simple or standard compound interest, which only compounds a set number of times per year. Our exercise takes advantage of this formula, showing that the accumulated amount after 7 years, with continuous compounding, is exactly three times the amount initially invested.