Problem 34 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The continuous growth rate equivalent to a 10% annual growth rate is approximately 9.531%.

Step 1: Understand the Problem

We are given an annual percent growth rate of 10% and need to find the equivalent continuous growth rate.

Step 2: Convert Percent to Decimal

An annual percent growth of 10% can be written as a decimal. This is done by dividing by 100, yielding 0.10.

Step 3: Use the Formula for Continuous Growth Rate

The formula for continuous growth rate is given by the relation between continuous and annual compounding, represented as: \[ r = \ln(1 + R) \]where \( R \) is the annual growth rate as a decimal.

Step 4: Compute the Natural Logarithm

Substitute 0.10 for \( R \) in the formula: \[ r = \ln(1 + 0.10) = \ln(1.10) \].Use a calculator to find the natural logarithm of 1.10.

Step 5: Calculate and Interpret the Result

Calculate \( \ln(1.10) \), which is approximately 0.09531. This represents the continuous growth rate as a decimal. Multiply by 100 to convert it into a percentage, resulting in approximately 9.531%.

Key Concepts

Annual Growth Rate

The annual growth rate measures how much something, like a population or investment, increases over a year. It's often expressed as a percentage and shows the average increase per year. If you have an annual growth rate of 10%, that means every year, the target quantity grows by 10% of its size from the previous year. Understanding this helps you anticipate what future values might look like.

• An example of its use can be seen in financial investments or population studies.
• In the context of investments, a 10% annual growth rate would mean a $100 investment grows to $110 over a year.

To compare growth rates over different periods or methods, like continuous growth (where the growth is constant at every instant), we often need to convert this annual rate into other forms.

Logarithms

Logarithms, particularly natural logarithms denoted as \(\ln\), are important mathematical tools. They are the inverse of exponentiation and help solve for growth rates that are applied continuously. With continuous growth, calculations can be tricky, so using natural logarithms simplifies the process.Natural logarithms work with the base \( e \), an irrational constant approximately equal to 2.71828. When converting between annual and continuous growth rates, natural logarithms help bridge the two different measurement scales.
For instance, in our step-by-step solution, the natural logarithm of 1.10, written as \( \ln(1.10) \), is computed to find an equivalent continuous growth rate. Calculating \( \ln(1.10) \) helps translate the 10% annual growth into a continuous context, demonstrating how much growth occurs if it's compounding continuously.

Growth Rate Conversion

Growth rate conversion involves changing the form of a growth rate to fit the context of analysis or comparison. In some cases, comparing annual growth to continuous growth requires conversion. Continuous growth rates assume the increase happens at every tiny moment, unlike annual rates which apply just once per year.
The conversion uses the formula:\[ r = \ln(1 + R)\]Here, \( R \) represents the annual growth rate in decimal form, and \( r \) is the continuous growth rate.

• First, convert the annual rate from a percentage to a decimal by dividing by 100.
• Then, compute the natural logarithm of \( 1 + R \). In our exercise, \( R = 0.10 \).
• Finally, this gives the continuous growth rate in decimal, which you can further convert into a percentage if needed by multiplying by 100.

This approach shows how a 10% annual growth translates into roughly a 9.531% continuous growth rate, illustrating how rates change with different compounding assumptions.