Problem 37 - Precalculus Mathematics for Calculus

Question

Investment \(A\) sum of \(\$ 5000\) is invested at an interest rate of \(9 \%\) per year, compounded continuously. (a) Find the value \(A(t)\) of the investment after \(t\) years. (b) Draw a graph of \(A(t).\) (c) Use the graph of \(A(t)\) to determine when this investment will amount to \(\$ 25,000.\)

Step-by-Step Solution

Answer

The investment will reach $25,000 after approximately 17.9 years.

Step 1: Understanding Continuous Compounding Formula

The formula used for continuous compounding is given by \[ A(t) = P \cdot e^{rt} \] where \( A(t) \) is the amount of money accumulated after \( t \) years, including interest, \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate (decimal), and \( t \) is the time in years.

Step 2: Define Variables and Calculate A(t)

Here, the principal amount \( P = 5000 \), the interest rate \( r = 0.09 \), and we need to find \( A(t) \). Substituting these into the formula:\[ A(t) = 5000 \cdot e^{0.09t} \] This is the formula for the value of the investment after \( t \) years.

Step 3: Graph A(t)

To graph \( A(t) = 5000 \cdot e^{0.09t} \), choose several values for \( t \) (e.g., \( t = 0, 5, 10, 15, 20 \)) and calculate corresponding \( A(t) \) values. Plot these points (\( t, A(t) \)) on a graph and draw the curve representing continuous growth over time.

Step 4: Determine Time When Investment Amounts to $25,000

Set \( A(t) \) to 25,000 and solve for \( t \): \[ 25000 = 5000 \cdot e^{0.09t} \]Divide both sides by 5000:\[ 5 = e^{0.09t} \]Take the natural logarithm of both sides:\[ \ln(5) = 0.09t \]Solve for \( t \):\[ t = \frac{\ln(5)}{0.09} \approx 17.895 \]So, the investment will amount to $25,000 in approximately 17.9 years.

Key Concepts

Continuous Compounding

Continuous compounding is a method used to calculate interest in which the frequency of compounding is maximized. Most investment accounts compound interest monthly or annually. Continuous compounding suggests that compounding occurs an infinite number of times, with the effect that at each moment, the interest earned is added back to the principal. This results in notable growth compared to traditional compounding methods.

For continuous compounding, we use the formula:

• \( A(t) = P \cdot e^{rt} \)

Here, \( e \) is the base of the natural logarithm, approximately equal to 2.71828. This method of compounding accelerates the growth of your investment by continuously applying interest at every instant, and is useful in understanding theoretical compound growth potential.

Natural Logarithm

Natural logarithms are a type of logarithm where the base is the constant \( e \). They are often used in mathematics and financial calculations involving exponential growth or decay due to their natural relation with continuous processes.

In our problem, when solving for the time \( t \) it takes for an investment to grow to $25,000, we make use of the natural logarithm. We reached a point where \( 5 = e^{0.09t} \). To find \( t \), we applied the natural logarithm to both sides:

• \( \ln(5) = 0.09t \)
• \( t = \frac{\ln(5)}{0.09} \)

This calculation showed us that \( t \approx 17.9 \) years. Natural logarithms provide a straightforward way to reverse exponential equations, which often arise in growth and interest calculations involving \( e \).

Investment Formula

The investment formula used for continuous compounding is central to calculating the future value of an investment over time. The formula \( A(t) = P \cdot e^{rt} \) is structured as follows:

• \( A(t) \): the future value of the investment after time \( t \).
• \( P \): the principal amount, or initial investment.
• \( r \): the annual nominal interest rate, in decimal form.
• \( t \): the time the money is invested for, in years.
• \( e \): a mathematical constant approximately equal to 2.71828.

In practice, this formula allows investors to understand how their contributions grow over time, particularly when interest is applied at every conceivable instant. For example, in our exercise, with \( P = 5000 \) and \( r = 0.09 \), we used this formula to determine how the investment grows over any given period \( t \). Such equations are crucial in financial planning and goal-setting.