Problem 39 - Applied Calculus

Question

What continuous percent growth rate is equivalent to an annual percent growth rate of \(10 \%\) ?

Step-by-Step Solution

Answer

A continuous growth rate of approximately 9.53% is equivalent to a 10% annual growth rate.

Step 1: Understand the Question

We need to find the continuous growth rate, usually denoted by the symbol \( r \), which gives equivalent growth over one year as an annual growth rate of \( 10\% \).

Step 2: Recall the Formulas

The formula for annual growth rate as a percent is \( (1 + r_a) = 1.10 \), where \( r_a = 0.10 \). For continuous growth, the formula is \( e^r \), where \( r \) is the continuous growth rate.

Step 3: Set the Equations Equal

To find the equivalent continuous growth rate, set the annual growth formula equal to the continuous growth formula: \[ e^r = 1.10 \]

Step 4: Solve for r

To solve for \( r \), take the natural logarithm \( \ln \) of both sides: \[ \ln(e^r) = \ln(1.10) \]Since \( \ln(e^r) = r \), it simplifies to: \[ r = \ln(1.10) \]

Step 5: Calculate the Natural Logarithm

Using a calculator, compute the natural logarithm: \[ r \approx \ln(1.10) \approx 0.0953 \]

Step 6: Convert to Percent

To convert \( r \) to a percent, multiply by 100: 0.0953 \( \times 100 \approx 9.53\% \).

Key Concepts

Exponential Growth

Exponential growth refers to the increase of a quantity by a constant percentage over a regular time interval. This type of growth is characterized by the formula \( P(t) = P_0 \, e^{rt} \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( e \) is the mathematical constant approximately equal to 2.71828, \( r \) is the continuous growth rate, and \( t \) is the time period.
This concept is widely applicable in natural processes such as population growth, radioactive decay, and financial investment. A key feature of exponential growth is that as the population increases, the amount of new growth becomes larger over the same time period. This makes it very powerful, but also can lead to unsustainable outcomes if not managed properly.
Understanding exponential growth helps you model and predict how things like cells, bacteria, or investments will grow over time, given a certain growth rate. This type of function curves upwards, showing an initially slow growth that speeds up over time. It's essential to understand how small changes in the growth rate \( r \) significantly influence the long-term growth outcome.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \), where \( e \) is approximately 2.71828. It is often referred to as "ln." The natural logarithm is the inverse function of exponential growth, specifically \( e^x \). This relationship means \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \).
The natural logarithm gives the time needed to reach a certain level of growth at a continuous rate. For example, in financial calculations, using \( \ln \) allows us to find the time over which continuously compounding interest would grow an investment to a desired amount.

• It is essential in solving equations where variables are in exponents, such as \( e^r = X \). Taking the \( \ln \) of both sides allows you to bring down the exponent and solve for \( r \).
• In the context of growth, it enables you to transform exponential growth equations into linear form, making them easier to analyze and solve.

The natural logarithm is particularly useful in various scientific and economic fields due to its constant base \( e \) and how it relates real-world data to exponential growth patterns.

Annual Growth Rate

The annual growth rate is a measure of how quickly an investment or other value increases over a year. It is usually expressed as a percentage, which indicates the amount of growth relative to the initial amount. The formula for calculating annual growth rate is given by \( (1 + r_a)^t = (1 + \, ext{annual growth rate}) \), where \( r_a \) is the annual growth rate in decimal form.
To convert the annual growth rate into a percent, you simply multiply \( r_a \) by 100. So if an investment grows by 10% annually, the formula becomes \( (1 + 0.10) \) or \( 1.10 \).
It's important to recognize that the annual growth rate does not account for compounding within the year, which is why sometimes continuous growth rates are used instead. These continuous rates offer a more precise and gradual representation of growth over time since they calculate growth continuously rather than in discrete time steps.

• Annual growth rates are helpful for quickly summarizing the expected rate of return or growth for a given year.
• They simplify financial planning by giving a straightforward percentage that can easily be communicated and understood.

Understanding the concept of annual growth and being able to convert it to continuous growth rates is crucial for accurate financial forecasting and economic analysis.