Problem 13 - Algebra 1: Concepts and Skills

Question

Write an exponential growth model for the profit. A business had a $10,000 profit in 1990. Then the profit increased by 25% per year for the next 10 years.

Step-by-Step Solution

Answer

The exponential growth model for the profit is \(P(t) = 10000 * e^{2.5}\)

Step 1: Identify the initial profit

The initial profit (\(P0\)) is given as $10,000

Step 2: Calculate the annual growth rate

The annual growth rate (\(r\)) is given as 25%. You need to convert this to decimal form by dividing by 100, resulting in 0.25

Step 3: Set the time frame

The time frame (\(t\)) is given as 10 years.

Step 4: Insert values into the exponential growth model

Substitute the values into the exponential growth equation \(P(t) = P0 * e^{rt}\), you get: \(P(t) = 10000 * e^{0.25*10}\)

Step 5: Simplify the equation

Simplify the equation to get: \(P(t) = 10000 * e^2.5\)

Key Concepts

Profit Calculation

When a business seeks to understand how its profits have grown or will grow over time, profit calculation becomes a crucial task. For this problem, starting with an initial profit is essential. The business began with $10,000 as profit in 1990.
This initial profit acts as the baseline to calculate further growth. Each year, this amount increased by a fixed percentage, which in this case is 25%. It's important to keep this initial figure in mind, as it sets the stage for applying mathematical models like the exponential growth equation to forecast future profits.
Understanding the starting point allows us to project growth adequately. However, it is essential to ensure that the growth figures are correctly interpreted, as the percentages build upon each successive year's profits.

Annual Growth Rate

The annual growth rate is a key element in determining how quickly profits are expected to rise over time. In the given problem, the growth rate is specified at 25% per year.
To use this in calculations, especially with exponential models, it's necessary to convert the percentage into a decimal form. This is done by dividing the percentage by 100. So, 25% becomes 0.25.

• This conversion is crucial for using growth rates in formulas.
• Distinguishing between percentages and decimals ensures accuracy in calculations.
• A higher annual growth rate indicates a faster increase in profits.

Converting and applying the growth rate correctly is essential to predicting the business's profit over the years accurately.

Exponential Growth Equation

The exponential growth equation is a fundamental tool in modeling situations where growth happens at a steady rate over time. In this scenario, the profits grow exponentially, meaning they increase by a fixed percentage each year.
The general form of the exponential growth equation is:\[ P(t) = P_0 \times e^{rt} \]where:

• \( P(t) \) is the profit at time \( t \).
• \( P_0 \) is the initial profit amount.
• \( r \) is the annual growth rate (expressed as a decimal).
• \( t \) is the time in years.

In the exercise, this was applied with an initial profit of $10,000, a growth rate of 25% (or 0.25 in decimal), over a span of 10 years, leading to:\[ P(t) = 10000 \times e^{2.5} \]Breaking it down step by step ensures that all parts of the equation fit together, providing a clear picture of expected profit growth. Understanding how each element interacts within the exponential growth equation helps visualize how a business's profits could evolve.