Problem 39 - Applied Calculus

Question

(a) Use the Rule of 70 to predict the doubling time of an investment which is earning \(8 \%\) interest per year. (b) Find the doubling time exactly, and compare your answer to part (a).

Step-by-Step Solution

Answer

(a) 8.75 years using the Rule of 70; (b) Exact doubling time is approximately 9.006 years.

Step 1: Using the Rule of 70

The Rule of 70 is a simple formula used to estimate the number of years required to double the size of an investment at a given annual growth rate. The formula is: \[ \text{Doubling Time} = \frac{70}{\text{Interest Rate}} \] Insert the given interest rate of 8% into the formula: \[ \text{Doubling Time} = \frac{70}{8} = 8.75 \text{ years} \]

Step 2: Calculate Doubling Time Exactly

To find the doubling time exactly, use the formula for exponential growth: \[ A = P(1 + r)^t \] where \(A = 2P\) (because we want to double the investment), \(r = 0.08\), and \(t\) is the doubling time. Substitute these into the equation: \[ 2 = (1 + 0.08)^t \] Take the natural logarithm of both sides: \[ \ln(2) = t \cdot \ln(1.08) \] Solve for \(t\): \[ t = \frac{\ln(2)}{\ln(1.08)} \approx \frac{0.693}{0.077} \approx 9.006 \text{ years} \]

Step 3: Compare the Results

The Rule of 70 gave us a doubling time of 8.75 years, while the exact calculation gave us approximately 9.006 years. The Rule of 70 provides a quick estimate that is relatively close to the exact calculation; however, it slightly underestimates the true doubling time.

Key Concepts

Doubling Time

Doubling time is a key concept when discussing exponential growth, particularly in finance and economics. It refers to the period it takes for an investment or quantity to double in size or value at a consistent rate of growth. One common way to approximate the doubling time is by using the Rule of 70. This rule provides a quick, straightforward formula: divide 70 by the annual growth rate (expressed as a percentage). For example, with an 8% interest rate, the doubling time is calculated as \( \frac{70}{8} = 8.75 \) years. This method, although handy for quick estimates, does have some limitations.However, to find the exact doubling time, more precise mathematical methods, such as exponential growth functions and natural logarithms, must be applied. Despite being slightly more complex, these methods ensure greater accuracy in determining when a quantity will truly double.

Exponential Growth

Exponential growth describes the process where quantities increase at a rate proportional to their current value. This results in growth that accelerates over time, manifesting as a curve that starts off relatively flat and becomes increasingly steep. In the context of financial investments, exponential growth is witnessed when interest is compounded, meaning the earned interest itself earns interest in subsequent periods.The formula for calculating future value with exponential growth is \( A = P(1 + r)^t \), where:

• \(A\) represents the total amount after time \(t\)
• \(P\) is the initial principal balance
• \(r\) is the annual interest rate
• \(t\) is the time in years

This function shows how even modest interest rates can lead to significant growth over time, evidencing why understanding exponential growth is crucial in various fields like finance, biology, and environmental science.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical function that is the inverse of exponentiation when the base is Euler's number \(e\) (approximately 2.71828). It is useful in calculating time-related aspects of exponential growth, as it helps to determine the time needed for a quantity to multiply a certain number of times.In finance, natural logarithms become particularly valuable in calculating exact doubling times. For example, to solve \( 2 = (1 + 0.08)^t \) for doubling time at an 8% interest rate, you take the natural logarithm of both sides to get \( \ln(2) = t \cdot \ln(1.08) \). Solving for \(t\) results in \( t = \frac{\ln(2)}{\ln(1.08)} \), providing a precise doubling time of approximately 9.006 years.This method ensures an accurate computation of time in scenarios involving continuous and compound growth, making understanding of natural logarithms essential across different scientific and financial applications.