Problem 18 - Algebra 1: Concepts and Skills

Question

Write an exponential model to represent the situation. Tell what each variable represents. A new sound system, valued at 800 dollar, decreases in value by \(10 \%\) each year.

Step-by-Step Solution

Answer

The exponential model representing this situation is \(P = 800e^{-0.1t}\), where \(P\) is the future value of the sound system, \(P_0 = $800\) is the initial value, \(r = 0.1\) (10%) is the yearly rate of depreciation, and \(t\) is time in years.

Step 1: Understanding the exponential decay model

Generally, an exponential decay model follows the rule \(P = P_0e^{-rt}\), where \(P\) is the future value of the object, \(P_0\) is the initial value, \(r\) is the rate at which the value is decreasing (our rate of decay), and \(t\) is the time period for which the value has been decreasing. We need to substitute our given values into this model.

Step 2: Substituting in the values

Here, \(P_0 = $800\), \(r = 10\%\) or \(0.1\) when converted to a decimal (since the value of the sound system decreases by \(10\%\) each year), and \(t\) is in years. We just plug these into the exponential decay equation. Now it looks like this: \(P = 800e^{-0.1t}\)

Step 3: Explanation of variables

In the context of this problem, \(P_0 = \$800\) is the initial value of the sound system, \(r = 0.1\) is the rate at which the sound system is decreasing in value (yearly), \(t\) represents time that has passed in years, and \(P\) represents the future value of the sound system after \(t\) years of decreasing by \(10\%\) each year.

Key Concepts

Exponential Functions in Decay Models

Exponential functions form the backbone of many mathematical models for processes that grow or decay over time. Exponential decay specifically describes situations where a quantity decreases at a rate proportional to its current value. The general formula for exponential decay is given by:

• \(P = P_0e^{-rt}\)

Here, \(P\) represents the future value after some time, \(P_0\) is the initial value, \(r\) is the rate of decay, and \(t\) is the time period. The key characteristic of an exponential function is that the rate of change of the function with respect to time is directly proportional to the value of the function at that time.
This allows such functions to precisely model scenarios like the depreciation of assets or the cooling of an object. Each component of the equation plays a crucial role in predicting future values based on the initial conditions and rate of change.

Understanding the Rate of Decay

The rate of decay is a critical factor in exponential decay models as it describes how quickly the value of an object is decreasing. In mathematical terms, it is represented by the constant \(r\) in the exponential decay equation. A higher rate of decay means the value decreases more rapidly over time, and is usually expressed as a percentage.

• For the sound system example, the rate of decay is 10% per year.
• This is converted to a decimal \(r = 0.1\) for use in the formula.

It's important to remember that a rate of decay signifies a proportional decrease. So, your future value diminishes by a percentage of its current value, not its original value every year. This often catches people by surprise.
To calculate this in practice, you continually apply the rate of decay year by year as time progresses. Recognizing how this works will help you apply similar principles to other scenarios showing decay.

The Role of Initial Value

Initial value, denoted as \(P_0\) in exponential functions, is the value at the beginning of your observation period. It serves as a starting point for calculation in models of growth or decay. In our scenario, the initial value is \$800, the price of the sound system before any depreciation.

• This initial figure anchors the exponential decay model.
• It is essential as the base from which you calculate subsequent decreases.

Every calculation of projected value stems from the initial value. As you apply the rate of decay to \(P_0\), you find new values depending on the duration, \(t\). Grasping the concept of the initial value helps in understanding the progression of value over time in any exponential decay situation.
Without the correct initial value, predicting future values accurately would not be possible.