Problem 69 - Precalculus : Building Concepts and Connections

Question

Find the interest rate \(r\) if the interest on the initial deposit is compounded continuously and no withdrawals or further deposits are made. Initial amount: \(\$ 4000 ;\) Amount in 8 years: \(\$ 6000\)

Step-by-Step Solution

Answer

The interest rate \(r\) is approximately equal to \(ln(1.5)/8\), or about 0.0513 per year.

Step 1 Title

First, substitute the given values into the formula \(A = Pe^{rt}\). This gives \(6000 = 4000e^{8r}\).

Step 2 Title

Next, we can manipulate this equation to solve for \(r\). First, divide both sides by 4000, yielding \(1.5 = e^{8r}\).

Step 3 Title

We then take the natural logarithm (ln) of both sides. The property of logarithms: ln(exp(a)) = a, allows us to bring down the exponent on the right-hand side. This gives \(ln(1.5) = 8r\).

Step 4 Title

Finally, solve for \(r\) by dividing both sides by 8. This gives \(r = ln(1.5)/8\).

Key Concepts

Continuous Compounding

When it comes to understanding money and finance, continuous compounding is an essential concept. Continuous compounding means that interest is calculated and added to the principal amount continuously, rather than at set intervals like annually, quarterly, or monthly. This concept is useful because it ensures that investment grows as much as possible in the shortest amount of time.
The formula for continuous compounding is:

• \( A = Pe^{rt} \)

Where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (the initial investment).
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
• \(r\) is the annual interest rate (in decimal form).
• \(t\) is the time the money is invested for.

Continuous compounding maximizes the growth of an investment or savings by recalculating interest on an ongoing basis. It ensures that the money works harder and grows faster than other compounding schedules.

Natural Logarithm

Natural logarithms are integral in solving equations where the variable is an exponent in a base of "e". The natural logarithm, often represented as \(\ln\), is the inverse operation to exponentiation with the base \(e\). This means that:\[\ln(e^x) = x\]
Natural logarithms help simplify equations in which exponential growth is involved. In the context of continuous compounding, if you have an equation like \(e^{8r} = 1.5\), to solve for \(r\), you take the natural logarithm of both sides, using the property \(\ln(e^x) = x\). By doing this, you can convert the exponential equation into a linear one, allowing for easier solution finding:

• \(\ln(1.5) = 8r\)

From there, you can continue solving for \(r\) by simple arithmetic. Hence, natural logarithms play a critical role in understanding and working with exponential functions, particularly in financial calculations.

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to its current value. This kind of growth results in the value increasing rapidly over time. In financial terms, exponential growth is significant because it describes scenarios where the value of money increases quickly due to compounding.
In the previous exercise, the amount of money grows from \\(4000 to \\)6000 over 8 years through continuous compounding, demonstrating exponential growth. This is modeled by the function \(A = Pe^{rt}\), where the exponential component \(e^{rt}\) signifies growth that accelerates as the principal grows. As shown in the formula, the effect of the interest rate and time is exponential because they appear as exponents.The concept of exponential growth is critical for understanding how investments can accumulate and grow over time, providing insights into how quickly an investment can expand under varying conditions. This understanding can significantly influence both personal and professional financial decision-making.