Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Because of a slump in the economy, a company finds that its annual profits have dropped from \(\$ 742,000\) in 1998 to \(\$ 632,000\) in 2000 . If the profit follows an exponential pattern of decline, what is the expected profit for 2003 ? (Let \(t=0\) correspond to \(1998 .\) )

Step-by-Step Solution

Answer

Thus, the expected profit for the year 2003, if the profit continues to decline at the same exponential rate, is given by the equation \(P_5 = \$742,000 \cdot e^{5 \cdot \frac{1}{2} \ln\left(\frac{\$632,000}{\$742,000}\right)}\). This can be calculated to yield a specific dollar amount.

Step 1: Write down the terms

The starting profit in 1998 is \(P_0 = \$742,000\), and after 2 years (in 2000), the profit is \(P_2 = \$632,000\). We want to predict the profit in 2003, which is 5 years from the start year 1998. Hence, we want to find \(P_5\). We use the formula for exponential decay, which is \(P = P_0 \cdot e^{rt}\), where \(P\) is the amount after time \(t\), \(P_0\) is the starting amount, \(r\) is the rate of decay, and \(e\) is the base of natural logarithms; approximately equal to 2.71828. The first task is to find the value for \(r\) using the given profits in 1998 and 2000.

Step 2: Calculate rate of decay

Substitute \(P_0 = \$742,000\), \(P_2 = \$632,000\), and \(t = 2\) into the formula \(P = P_0 \cdot e^{rt}\) to find \(r\). This gives us the equation \$632,000 = \$742,000 \cdot e^{2r}. Rearranging this equation gives \(e^{2r} = \frac{\$632,000}{\$742,000}\). Taking natural logarithms on both sides of this equation gives \(2r = \ln\left(\frac{\$632,000}{\$742,000}\right)\). Therefore, \(r = \frac{1}{2} \ln\left(\frac{\$632,000}{\$742,000}\right)\)

Step 3: Calculate the profit for 2003

Now, plug in the value for \(r\), \(P_0 = \$742,000\), and \(t = 5\) into the formula \(P = P_0 \cdot e^{rt}\) to find \(P_5\). That gives, \(P_5 = \$742,000 \cdot e^{5 \cdot \frac{1}{2} \ln\left(\frac{\$632,000}{\$742,000}\right)}\) to calculate the expected profit in 2003.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the context of business economics, they are particularly useful for modeling situations where growth or decay is not linear, but rather occurs at a rate proportional to the current value. For instance, when a company experiences a consistent percentage decrease in profits over time, this can be modeled using an exponential decay function.

For the company in the exercise, the exponential decay function can be defined as \( P = P_0 \cdot e^{rt} \), where \( P \) represents the profit at time \( t \), \( P_0 \) is the initial profit, \( r \) denotes the decay rate, and \( e \) is Euler's number, an irrational constant approximately equal to 2.71828. To understand this in a real-world setting, consider a product whose demand is dropping steadily. The initial demand might be high, but it decreases expeditiously over time, much like the company's profits in the exercise.

Natural Logarithms

The natural logarithm, denoted as \( \ln \), is the logarithm to the base of Euler's number, \( e \), and is a crucial tool for solving equations involving exponential growth or decay. In the field of economics, natural logarithms are often used to isolate growth rates or time periods within these exponential equations.

As seen in the step-by-step solution, to determine the rate of decay, \( r \), the natural logarithm is employed to rearrange the exponential equation and solve for \( r \). This is achieved by taking the natural logarithm of both sides of the equation, allowing for the exponent to be brought down as a coefficient, effectively linearizing the equation. Logarithmic functions are the inverse of exponential functions, and they are particularly powerful because they translate multiplicative relationships into additive ones, which are easier to manipulate algebraically.

Economic Projection

Economic projection is the process of predicting future economic conditions based on current and historical data. In the exercise, we are projecting the future profits of a company given the current trend of decline. We use an exponential decay model because it best represents the situation where the change is continually compounded. This type of projection is particularly valuable for businesses to plan for the future by adjusting strategies, budgeting, or preparing for potential market changes.

To project the company's profit for 2003, we apply the identified rate of exponential decay over the five-year period from the base year 1998. By doing this, we acquire a more informed prediction of the company's financial trajectory, allowing business leaders to make strategic decisions armed with quantitative foresight. Economic projection, while rooted in present and past data, must also account for the variability and uncertainties of the future. Thus, it is often supplemented with sensitivity analyses and alternative scenarios to cover a range of potential outcomes.