Problem 26 - Applied Calculus

Question

The company that produces Cliffs Notes (abridged versions of classic literature) was started in 1958 with \(\$ 4000\) and sold in 1998 for \(\$ 14,000,000 .\) Find the annual percent increase in the value of this company over the 40 years.

Step-by-Step Solution

Answer

The annual percent increase is approximately 19.18%.

Step 1: Understand the Problem

The company started in 1958 with an initial value of \( P_0 = \\(4000 \) and was sold in 1998 for \( P_{40} = \\)14,000,000 \). We need to find the annual percent increase over 40 years.

Step 2: Identify the Appropriate Formula

We can use the formula for exponential growth, \[P = P_0(1 + r)^t\]where \(P\) is the future value, \(P_0\) is the initial value, \(r\) is the annual growth rate (as a decimal), and \(t\) is the time in years.

Step 3: Substitute Known Values Into The Formula

We substitute \( P_{40} = 14,000,000 \), \( P_0 = 4000 \), and \( t = 40 \) into the formula:\[14,000,000 = 4,000(1 + r)^{40}\]

Step 4: Solve for \( 1 + r \)

First, divide both sides by 4000 to isolate the exponential term:\[\frac{14,000,000}{4,000} = (1 + r)^{40}\]Simplifying, we get:\[3500 = (1 + r)^{40}\]

Step 5: Solve for \( r \)

To find \( r \), take the 40th root of both sides:\[1 + r = 3500^{\frac{1}{40}}\]Calculate this, and then subtract 1 to find the annual growth rate \( r \).

Step 6: Convert \( r \) to a Percentage

Finally, multiply \( r \) by 100 to convert it to a percentage, the annual percent increase.

Key Concepts

Annual Percent Increase

When you hear about an annual percent increase, you might wonder what it really means for a company or investment. It's a measure of how much a value grows each year, expressed as a percentage. This helps in understanding growth over time. Let's see how it applies to a real-world scenario.

For instance, consider a company that was valued at \(4,000 in 1958 and became worth \)14,000,000 in 1998. To figure out the annual percent increase over those 40 years, consider the initial and final values.

• Initial Value, \( P_0 \): The starting point, which was \(4,000.
• Final Value, \( P_{40} \): The value after 40 years, which was \)14,000,000.
• Time, \( t \): The period over which the growth occurred, which was 40 years.

The annual percent increase provides insight into the consistent growth rate of an investment or value after every year in this period.

Calculating this involves determining a growth rate, then translating it into a percentage format to make it more understandable.

Future Value Calculation

Calculating future value is an important part of understanding how your initial investment or value grows over time. In the context of exponential growth, future value is depicted as \( P \) in our formula.

The formula used is:\[P = P_0(1 + r)^t\]Where:

• \( P \): Final or future value (what the investment is worth after time \( t \))
• \( P_0 \): Initial value (the starting amount, e.g., \(4,000)
• \( r \): Annual growth rate (as a decimal)
• \( t \): The number of years the value has grown

By determining future value, we can predict how much an investment will grow. For example, using the known values in the scenario, the future value helps see the growth from \)4,000 to $14,000,000 in 40 years.

This calculation gives insights into long-term growth potential and helps make informed financial decisions.

Exponential Function

Exponential functions are crucial in understanding growth processes like interest or population increase. They are characterized by a constant growth rate expressed as a percentage.

The basic formula, \( P = P_0(1 + r)^t \), models this growth. The key components of exponential growth include:

• Initial Value \( (P_0) \): The starting amount of the entity being measured, like $4,000 in our example.
• Growth Factor (\( 1 + r \)): Represents the growth one year at a time, with \( r \) being the annual rate as a decimal.
• Exponent \( (t) \): The period over which the growth is observed, like 40 years in our scenario.

In theory, as long as the growth is consistent year over year, this function can show how initial amounts multiply exponentially, producing results significantly larger than the inputs.

This power of exponential functions amplifies over longer durations, hence making it a powerful tool to project growth, allowing us to turn complex, compounded growth into simple calculations.