Question

EXPONENTIAL MODELS Tell whether the situation can be represented by a model of exponential growth or exponential decay. Then write a model that represents the situation. (Review \(8.5,8.6)\) PINTERNET SERVICE From 1993 to \(1998,\) the total revenues for a company that provides Internet service increased by about \(137 \%\) per year.

Short answer

The situation experiences exponential growth. The model that represents the situation is \( P(t) = P_0*(2.37)^t \).

Step-by-step solution

Understand the Problem

From 1993 to 1998, the total revenues for an Internet service company increased by about 137% per year. That information hints towards an exponential growth problem.

Understand Exponential Growth

In Exponential growth, a quantity increases exponentially over a period of time, in this case by 137% annually. Therefore, the representation of this growth can be formulated as \( P(t) = P_0*(1 + r)^t \), where \( P(t) \) is the future value of the investment, \( P_0 \) is the initial amount, \( r \) is the annual growth rate (expressed as a decimal), and \( t \) is the time in years.

Formulate the Equation

We are not provided with the original amount of revenue, so we can represent it generically as P0. The annual growth rate is 137% read as 137/100 or 1.37. Our equation would then look like this: \( P(t) = P_0*(1 + 1.37)^t \).

Key concepts

Exponential Decay

Exponential decay describes the process by which a quantity diminishes at a rate proportional to its current value. This concept is often used in the contexts of radioactive decay, depreciation of assets, and population decline in a constrained environment.

In contrast to exponential growth, exponential decay models involve a percent decrease over time. Mathematically, the model for exponential decay is similar to growth, but it features a negative growth rate. For instance, if a certain substance has a half-life (the time it takes for half of the material to decay) of 5 years, the decay rate can be represented as a 50% decrease every 5 years. The equation for exponential decay is written as:
\[ P(t) = P_0 * (1 - r)^t \]
Where:

• \( P(t) \) is the amount remaining at time \( t \).
• \( P_0 \) is the initial amount of the substance.
• \( r \) is the decay rate expressed as a decimal.
• \( t \) is the time elapsed.

For effective learning, students should work on problems involving both exponential growth and decay to compare and contrast the behaviors of these models.

Formulate Exponential Equations

Formulating exponential equations involves translating a real-world situation into an exponential model that captures the essence of exponential change—whether growth or decay. To construct such an equation, four main components must be identified: the initial quantity \( P_0 \), the rate of change \( r \), the time variable \( t \), and the final quantity \( P(t) \).

Let's break the process down:

Identify Initial Amount (\( P_0 \))

Begin by determining the starting value of the quantity under consideration. This figure is crucial, as it serves as the baseline from which all change is measured.

Determine the Rate of Change (\( r \))

Next, establish whether the quantity is increasing or decreasing over time, and at what percentage rate. Remember, growth rate should be positive, while decay rate is negative. Convert the percentage into a decimal for calculation purposes.

Choose the Time Factor (\( t \))

This component tracks the duration over which the change occurs. It could be years, months, days, or any other unit of time, depending on the scenario.

Formulate the Equation

Finally, using the formula \( P(t) = P_0*(1 \pm r)^t \), you can craft the exponential equation. The positive sign is used for growth, and the negative sign for decay, as mentioned earlier.
Students should practice setting up and solving these equations with both theoretical and practical examples to internalize the steps involved.

Percent Increase in Exponential Growth

Percent increase in exponential growth refers to the rate at which a quantity is escalating over time in a manner that the change is proportional to the current value. This type of growth is constant in percentage, not in absolute terms, leading to larger increases as the quantity grows.

In the context of our Internet service company example, a growth of 137% per year implies that each year's revenue is 137% greater than the previous year's revenue. It's essential to note that the percent increase, when converted into a growth rate for an equation, is written in its decimal form. Thus a 137% increase translates to a growth rate \( r \) of 1.37 in the model equation.
In an exponential growth model, the percent increase can be calculated over time using the formula:\[ P(t) = P_0 * (1 + r)^t \]Where \( 1 + r \) symbolizes the total growth factor per unit of time. Here, the initial amount \( P_0 \) gets multiplied by this growth factor raised to the power of the number of time periods \( t \), accounting for the compound nature of exponential growth.
It is crucial to distinguish between simple interest (linear growth) and exponential growth—the latter will drastically outpace linear growth over time due to compounding. Exercises in determining the percent increase of exponential growth can help students better understand the power of compounding over time, which is a fundamental concept in finance and natural sciences.