Problem 22 - Algebra 1

Question

Write an exponential growth model. A business had a \(\$ 20,000\) profit in \(1990 .\) Then the profit increased by \(20 \%\) per year for the next 10 years.

Step-by-Step Solution

Answer

The exponential growth model for the business is \(P(t) = \$20,000 \cdot 1.2^t\), where \(P(t)\) represents the profit in year \(t\) after 1990.

Step 1: Identify the components of the exponential growth formula

The components of the exponential growth formula for this task are as follows: \(P_0 = \$20,000\), which is the initial profit; \(r = 0.2\) (20% converted into a decimal); and \(t\), which represents the number of years and would be different for different years.

Step 2: Plug the known values into the growth formula

The exponential growth model for this business is represented using these values in the formula: \(P(t) = \$20,000 \cdot (1 + 0.2)^t\). This formula calculates the profit in any given year after 1990.

Step 3: Simplify the formula

Simplify the exponential growth model further: \(P(t) = \$20,000 \cdot 1.2^t\). This model can now be used to calculate the business's profit in any given year.

Key Concepts

initial profit

When we talk about initial profit in the context of an exponential growth model, we are referring to the profit value from which growth begins. Initial profit is sometimes called the baseline value in a financial scenario. For a business, this represents the profit at the very start of the observation period, before any growth has occurred.

In the example of the business that made a profit in 1990, the initial profit is clearly stated. The business started with an initial profit of \( \$20,000 \). This figure is crucial as it serves as the starting point for calculating profits in subsequent years using the exponential growth model.

Understanding initial profit is important because it provides a clear reference point. All subsequent profits over the years are evaluated in relation to this starting amount. Without it, you cannot accurately determine how much the profit has grown with time.

growth rate

The growth rate in an exponential growth context reflects how quickly the profit is increasing per year. This rate is expressed as a percentage and is key to understanding how the initial profit will change over time.

In the given example, the business had a growth rate of 20% per year. Converting this percentage into a decimal is crucial for calculations. Therefore, 20% becomes \( r = 0.2 \). This transformation makes it easier to work with the exponential growth formula.

Knowing the growth rate helps predict future profits and plan for long-term growth. It reflects the speed of development in a business, providing valuable insights into its financial health. Whether this growth rate is sustainable and how it compares to similar businesses is another critical aspect of financial analysis.

exponential growth formula

The exponential growth formula is a mathematical expression used to calculate the growth of a quantity over time. It takes into account the initial starting value, the growth rate, and the period over which growth occurs. The general form of the exponential growth formula is:

\[ P(t) = P_0 \cdot (1 + r)^t \]

Where:

• \( P(t) \) is the profit at time \( t \)
• \( P_0 \) is the initial profit
• \( r \) is the growth rate
• \( t \) is the time in years

In our example, the formula becomes \( P(t) = \$20,000 \cdot (1.2)^t \).

This formula allows businesses to forecast future profits. By plugging different values of \( t \), a business can find out its expected profit for any year moving forward from 1990. As depicted, it simplifies the tracking process over time, ensuring businesses can make informed financial decisions.