Problem 17 - Applied Calculus

Question

A firm decides to increase output at a constant relative rate from its current level of 20,000 to 30,000 units during the next five years. Calculate the annual percent rate of increase required to achieve this growth.

Step-by-Step Solution

Answer

The required annual percent rate of increase is approximately 8.45%.

Step 1: Understand the Problem

We are given the current output of a firm, which is 20,000 units, and the target output after 5 years, which is 30,000 units. The goal is to find the consistent annual percentage growth rate that achieves this increase.

Step 2: Set Up the Exponential Growth Formula

The exponential growth formula can help us determine the required growth rate: \[ P(t) = P_0 (1 + r)^t \]where \( P_0 \) is the initial amount (20,000 units), \( P(t) \) is the future amount (30,000 units), \( r \) is the growth rate, and \( t \) is time in years (5 years in this case).

Step 3: Substitute Known Values into the Formula

Substitute the given values into the formula:\[ 30,000 = 20,000 (1 + r)^5 \]

Step 4: Solve for the Growth Rate \( r \)

To find \( r \), first divide both sides by 20,000:\[ 1.5 = (1 + r)^5 \]Now, take the fifth root of both sides:\[ (1 + r) = (1.5)^{1/5} \]

Step 5: Calculate the Growth Rate

Now, calculate \((1.5)^{1/5}\) using a calculator:\[ (1.5)^{1/5} \approx 1.08447 \]Therefore, \( 1 + r \approx 1.08447 \).To find \( r \), subtract 1 from both sides:\[ r \approx 0.08447 \]

Step 6: Convert the Growth Rate to a Percentage

Convert the growth rate to a percentage by multiplying by 100:\[ r \approx 0.08447 \times 100 \approx 8.447\% \]

Key Concepts

Annual Growth Rate

The concept of annual growth rate is critical to understanding how quantities increase over time. In the context of business or finance, it allows us to measure the increase in a firm's production, revenue, or other key metrics year over year.

When we talk about an "annual growth rate," we're referring to the percentage increase that occurs each year. This rate is consistent annually, meaning that the same percentage increase applies every year over the period being considered. It provides a straightforward way to understand how something grows from a starting point to a target value over time. For example, if a firm wants to increase its production from 20,000 units to 30,000 units over five years, understanding the required annual growth rate is essential for planning resources and strategy.

To find the annual growth rate, one must first identify the period of growth and the initial and final values. This rate is often expressed as a percentage, which makes it easier to interpret and apply in real-world situations. It's a pivotal figure in making business decisions related to expansion, investment, and future planning.

Exponential Growth Formula

The exponential growth formula is a powerful tool used to calculate growth over a specified period when the growth rate is constant. This formula is especially valuable in scenarios where we anticipate growth that compounds annually. The formula is expressed as:
\[ P(t) = P_0 (1 + r)^t \]
Here's a breakdown of the components:

• \( P(t) \): the future value of the quantity after time \( t \)
  
• \( P_0 \): the initial value or starting amount
  
• \( r \): the annual growth rate, expressed as a decimal
  
• \( t \): the number of years over which the growth occurs
  

This formula calculates how much a start value will grow after a certain number of years, given a constant growth rate. In our example, the firm needs to understand how much its production will increase each year, compounded over five years until it reaches the target value of 30,000 units from an initial output of 20,000 units. By setting the future value \( P(t) \) to 30,000 and initial value \( P_0 \) to 20,000, and solving for \( r \), we determine the consistent annual increase rate.

Percentage Growth Rate

The percentage growth rate is a way to express changes in numerical data over time. This rate essentially outlines how much a value has increased as a percentage of its initial value, over a certain period. Calculating it involves transforming a decimal growth factor into a percentage by multiplying by 100.

In the context of the exercise, once the growth factor \( (1 + r) \) was determined through the exponential growth formula, converting it to a percentage gives us the growth rate in percentage terms. For instance, if our growth factor \( (1 + r) \) came out to be approximately 1.08447, the percentage growth rate becomes:

• Subtract 1 from the growth factor to get \( r \), which is 0.08447.
  
• Convert \( r \) to a percentage: \( 0.08447 \times 100 \approx 8.447\% \)
  

This percentage makes it easy to discuss and compare growth rates in a manner that's simple to understand and apply. It's an essential aspect of strategic planning and forecasting, helping firms and individuals make informed decisions based on historical and projected growth patterns.