Problem 61 - Precalculus : Building Concepts and Connections

Question

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 1500 ; r=6 \%\)

Step-by-Step Solution

Answer

The given investment will take about 11.55 years to double when compounded continuously at an interest rate of 6%.

Step 1: Insert the given variables into the formula

The initial investment is $1500 and the rate is 6%, which we express as 0.06 in decimal form. If the investment doubles, the final amount is $3000. According to the formula for continuously compounded interest \( A = P e^{rt} \), the values are replaced as \( 3000 = 1500 \cdot e^{0.06t} \).

Step 2: Simplify the equation

In order to find \( t \), divide both sides of the equation by 1500, which results in \( 2 = e^{0.06t} \)

Step 3: Find the value of \( t \)

Now the purpose is to get \( t \) isolated on one side of the equation. Start by taking natural logarithm (ln) on both sides of the equation to get rid of \( e \). This results in \( \ln2 = \ln(e^{0.06t} ) \), which can be simplified (using the properties of logarithms) to \( \ln2 = 0.06t \).

Step 4: Solve for \( t \)

Finally, to get the value of \( t \), divide both sides of the equation by 0.06. Thus, \( t = \frac{\ln2}{0.06} \), approximately equals to 11.55 years.

Key Concepts

Exponential Growth

Exponential growth refers to the increase of an amount at a rate proportional to the value of the amount itself. In the context of finance and investments, this concept is particularly relevant when dealing with interest that is compounded continuously. Unlike simple or compound interest, which are calculated at discrete intervals, continuously compounded interest grows exponentially, accruing every possible instant.

Mathematically, exponential growth is modeled by the function A = Pe^{rt}, where A is the final amount, P is the principal amount, r is the annual interest rate (expressed as a decimal), and t is time in years. The constant e is Euler's number, approximately equal to 2.71828, which serves as the base for the natural logarithm. This constant is fundamental in many areas of mathematics and plays a crucial role in the growth formula.

Investments growing at a rate of r will increase more substantially over time compared to those that receive simple interest. This is because each increment of interest also begins to earn interest, leading to a snowball effect.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, which is Euler's number. This function is inverse to the exponential function with base e, meaning that ln(e^x) = x and e^{ln(x)} = x. In finance, the natural logarithm becomes essential when solving for variables in exponential growth equations, especially when we deal with continuously compounded interest.

In the given solution step, the natural logarithm was employed to isolate t, the time variable, in the equation 2 = e^{0.06t}. By applying ln to both sides, we obtained ln(2) = 0.06t, which allows us to solve for t directly. Without the natural logarithm, it would be difficult to manipulate the exponential equation to find the value of time needed for an investment to double.

Time to Double Investment

Determining the time it takes for an investment to double is a common financial calculation, often referred to as the 'Rule of 72' in its simpler form. However, when dealing with continuously compounded interest, the precise method involves using the natural logarithm.

To find the doubling time, we set up the equation 2 = e^{rt}, where 2 represents the doubling of the initial investment. By taking the natural logarithm of both sides, we can solve for t and find that t = (ln(2)) / r. This equation gives a more accurate result than the Rule of 72, especially when the interest is compounded continuously and the rates are higher.

Working through the provided example, with a 6% interest rate, the investment doubles in approximately 11.55 years. Understanding this concept helps investors and savers to project the growth of their funds over time based on the continuous compounding of interest.