Problem 21 - Algebra 1

Question

Write an exponential growth model. 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 given situation is \( P(t) = 10,000 * (1 + 0.25)^t \).

Step 1: Identify Known Values

First identify the known values from the problem: \( P_0 = \$10,000 \), \( r = 0.25 \) (which is \25\% expressed as a decimal), and \( t = 10 \) years.

Step 2: Structure the Exponential Growth Model

Next, understand that the exponential growth model takes the form \( P(t) = P_0 * (1 + r)^t \).

Step 3: Formulate the Exponential Model

Finally, substitute the known values into the exponential growth model. This yields the exponential growth model for the given situation: \( P(t) = 10,000 * (1 + 0.25)^t \).

Key Concepts

Profit Increase

Profit increase refers to the consistent growth of a business’s earnings over time. In the context of our example, the business experienced a profit increase of 25% each year for 10 years. This means that every year, the profit grows by a quarter of the previous year's profit. So, if the original profit was \( \\(10,000 \), the next year's profit would be 25% more. That is \( 10,000 \times 0.25 = \\)2,500 \) more, making the profit for that year \( \$12,500 \).

• Understanding percentage increases: When a profit increases by a percentage, it adds that percentage of the current amount to itself.
• Annual profit increase: This forms a pattern where each year's profit is calculated by adding 25% to the last year's profit.
• Impact: Compound growth results in profits that grow exponentially rather than just linearly.

This steady increase is what makes exponential growth models effective in capturing real-world growth scenarios.

Mathematical Modeling

Mathematical modeling involves creating a representation of a real-world situation using mathematical concepts. For the business profit scenario, an exponential model is appropriate because profits grow by a constant percentage each year. The formula used is \( P(t) = P_0 \times (1 + r)^t \), where:

• \( P(t) \) represents the profit after a certain number of years, \( t \).
• \( P_0 \) is the initial amount (the profit at year zero, or 1990 in this case).
• \( r \) is the rate of growth expressed as a decimal (25% becomes 0.25).
• \( t \) is the time in years.

By substituting the initial values and the rate of increase into the formula, we create a model that forecasts future profits. Mathematical models like these help businesses predict financial outcomes and plan for sustainable growth. They're crucial for strategic decision-making as they provide insights into future profitability.

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In our profit increase model, the exponential function is utilized to illustrate the rapid growth in profits over time. The structure of an exponential function pertinent to this context is \( (1 + r)^t \), where:

• The base \( (1 + r) \) is 1 plus the growth rate, indicating that the current year's profit is compounded by the rate of growth.
• The exponent \( t \) represents time in years, demonstrating the power of compounding over a decade.

This function elegantly captures the way small, consistent increases lead to large overall changes when applied repeatedly over several periods. For instance, after substituting the known values \( r = 0.25 \) and \( t \) ranging from 1 to 10, the model shows how profits grow significantly due to the compounding effect inherent in exponential functions.Exponential functions are pivotal in various domains like finance, population growth analysis, and even in technological forecasting, due to their ability to model persistent change efficiently.