Problem 23 - A Graphical Approach to College Algebra

Question

Suppose that \(\$ 2500\) is invested in an account that pays interest compounded continuously. Find the amount of time that it would take for the account to grow to the given amount at the given rate of interest. Round to the nearest tenth of a year. \(\$ 5000\) at \(5 \%\)

Step-by-Step Solution

Answer

It takes approximately 13.9 years for the investment to grow to $5000.

Step 1: Identify the Formula for Continuous Compound Interest

The formula for continuous compound interest is \[ A = Pe^{rt} \]where \( A \) is the amount of money accumulated after \( t \) years, \( P \) is the principal amount (initial investment), \( r \) is the interest rate, and \( e \) is the base of the natural logarithm.

Step 2: Substitute the Known Values

Given that the principal \( P \) is \\(2500, the interest rate \( r \) is \( 0.05 \) (or \( 5\% \)), and the future amount \( A \) is \\)5000, substitute these values into the formula: \[ 5000 = 2500 \times e^{0.05t} \]

Step 3: Solve for Time \( t \)

First, divide both sides of the equation by \( 2500 \) to isolate the exponential function:\[ 2 = e^{0.05t} \]Next, take the natural logarithm of both sides to solve for \( t \):\[ \ln(2) = 0.05t \]

Step 4: Calculate \( t \)

Divide both sides by \( 0.05 \) to solve for \( t \): \[ t = \frac{\ln(2)}{0.05} \]

Step 5: Compute the Value

Using a calculator, find \( \ln(2) \approx 0.6931 \). Then:\[ t = \frac{0.6931}{0.05} \approx 13.862 \]So, the time it takes for the money to grow to \$5000 is approximately \( 13.9 \) years, when rounded to the nearest tenth.

Key Concepts

Exponential Growth

Exponential growth is best explained as a process where the quantity increases rapidly over time, proportional to the current amount. In financial settings, this is seen in investments or savings accounts where interest accrues continuously. Continuous compounding occurs when interest is calculated on an infinitely small time interval, leading to growth that follows an exponential curve.

In the formula for continuous compound interest, \( A = Pe^{rt} \), the exponential function \( e^{rt} \) dictates how fast the principal \( P \) grows to the amount \( A \). The constant \( e \), known as Euler's number (approximately 2.71828), is key in modeling such exponential processes. Understanding how \( e \) affects growth is crucial in financial calculations, particularly in comparing different growth scenarios tied to varying interest rates and compounding strategies. This formula highlights the exponential relationship, illustrating that as time \( t \) increases, the effects on growth become more pronounced.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the inverse operation of the exponential function \( e^x \). It helps to "undo" the exponentiation process. This is particularly useful when trying to solve for variables positioned as exponents in equations.

In the scenario of our exercise, determining the time \( t \) required for an investment to double requires isolating \( t \) from the equation \( 2 = e^{0.05t} \). By applying the natural logarithm to both sides, we obtain \( \ln(2) = 0.05t \), effectively removing the \( e \) and simplifying the process of solving for \( t \).

The natural logarithm has unique properties, such as \( \ln(e) = 1 \) and \( \ln(1) = 0 \), which can be very helpful in simplifying expressions and solving equations, both in theoretical and real-world contexts.

Interest Rate Calculation

Interest rate calculation is pivotal in predicting how investments will perform over time. It involves understanding the rate at which money grows when invested. Often expressed as a percentage, the interest rate informs us how much of a given amount returns periodically.

In continuous compounding, the interest rate impacts the exponent in the equation \( e^{rt} \). Higher rates result in steeper exponential growth curves, leading to faster increases in investment value. Conversely, lower rates yield gentler curves, indicating slower growth. In our exercise, a 5% interest rate was applied.

An effective understanding of how to calculate the interest rate and its implications on growth is invaluable for making informed financial decisions. By examining different interest scenarios, investors can choose strategies that align with their financial goals, estimating future value and required investment periods.