Problem 69 - Algebra 2

Question

An investment of \(\$ 750\) will be worth \(\$ 1500\) after 12 years of continuous compounding at a fixed interest rate. What is that interest rate? $$ \begin{array}{lllll}{\text { A. } 2.00 \%} & {\text { B. } 5.78 \%} & {\text { C. } 6.93 \%} & {\text { D. } 200 \%}\end{array} $$

Step-by-Step Solution

Answer

The interest rate is approximately 5.78%

Step 1: Understanding and rearranging the continuous compounding formula

The formula for continuous compounding is \(A = Pe^{rt}\), where A is the ending balance, P is the initial investment (or principal), r is the annual interest rate and t is the time in years. In order to find the interest rate r, the formula needs to be rearranged into: \[r =\frac{1}{t} ln(\frac{A}{P})\] where ln is the symbol for natural logarithm.

Step 2: Substituting values

Substitute the given values into the rearranged formula. The initial investment P is $750, the ending balance A is $1500 and the time t is 12 years. This gives: \[r = \frac{1}{12} ln(\frac{1500}{750})\].

Step 3: Calculating the interest rate

Calculating the value inside the natural log gives 2, and plugging this value into the formula gives: \[r = \frac{1}{12} ln(2)\]. Now compute the value of the natural log of 2, then divide by the time t (12). Then, multiply by 100 to get the percentage.

Key Concepts

Interest Rate Calculation

Calculating the interest rate is vital when dealing with investments, particularly when they involve continuous compounding. The interest rate represents the percentage at which money grows over time. In this context, we're using a specific formula to determine the rate at which an initial investment grows continuously.

In continuous compounding, the formula used is \[A = P e^{rt},\]where:

• \(A\) is the future value of the investment (in this case, \\(1500),
• \(P\) is the principal amount or the initial investment (\\)750),
• \(r\) is the annual interest rate we want to find, and
• \(t\) is the time period in years (12 years here).

To extract the interest rate \(r\), we rearrange the formula to:\[r = \frac{1}{t} \ln \left(\frac{A}{P}\right),\]allowing us to calculate \(r\) directly using known values. Hence, understanding how to dissect these elements and calculate the rate is key to managing investments effectively.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is essential in the process of calculating continuous compounding interest rates. The natural logarithm is a mathematical concept that represents the power to which the base of the natural log (Euler's number, \(e\)) must be raised to produce a given number.

In our problem, after rearranging the continuous compounding formula, we have\[r = \frac{1}{t} \ln \left(\frac{A}{P}\right).\]This expression uses the natural logarithm to determine how many times you need to multiply \(e\) to turn the ratio \(\frac{A}{P}\) (which is 2 in our situation, since \(\frac{1500}{750} = 2\)) into its exponential form.

Understanding and calculating \(\ln(2)\) correctly is crucial as it directly influences the accuracy of our interest rate calculation. Remember, using a calculator or software that correctly computes the natural logarithm is often necessary.

Exponential Growth Formula

The exponential growth formula is a mathematical expression used to calculate how an investment grows continuously over time. This formula is expressed as:\[A = P e^{rt}\]and it models the continuously compounding interest scenario.

Let's break down the parts:

• \(A\) is the accumulated amount after time \(t\), including interest.
• \(P\) is the principal amount — the initial sum of money invested.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828, a constant in mathematics indicating continuous growth.
• \(r\) is the annual interest rate.
• \(t\) is the time in years that the money is invested or borrowed for.

This formula illustrates how even a small interest rate can result in significant growth over a long period due to the power of exponential compounding. In our example, calculating \(r\) gives us insight into how quickly an investment doubles, demonstrating the powerful effect of continuous compounding.