Problem 16 - Algebra 1: Concepts and Skills

Question

Write an exponential model to represent the situation. Tell what each variable represents. A 25,000 dollar car depreciates at a rate of \(9 \%\) each year.

Step-by-Step Solution

Answer

The exponential model representing the situation is \(P(t) = 25000 * (0.91)^t\). This model gives the car's value \(P(t)\) after a given number of years \(t\), where it decreases by \(9\%\) annually. The initial value \(P_0\) of the car is $25,000.

Step 1: Identify the initial value

The initial value of the car, or the value at time \(t = 0\), is $25,000. This will be the base value in our exponential equation. We'll denote it as \(P_0\).

Step 2: Identify the rate of change

The car depreciates at a rate of \(9\%\) each year. We need to express this as a decimal for our calculations, so \(9\% = 0.09\). But, this is a case of depreciation so it will be represented as a reduction from the whole, that is, \(1 - 0.09 = 0.91\) or \(91\%\). We'll denote this rate of change as \(r\).

Step 3: Write the exponential model

An exponential model can be written in the form \(P(t) = P_0 * (1 + r)^t\). Substituting our values in, we get the model: \(P(t) = 25000 * (0.91)^t\). Here, \(P(t)\) represents the car's value after \(t\) years, \(P_0\) is the car's initial value, \(r\) is the rate at which it depreciates annually, and \(t\) represents the time (in years).

Key Concepts

Depreciation

When we talk about depreciation, we are referring to a decrease in the value of an asset over time. In this case, it's about how a car loses its value each year. You can think of depreciation like the natural wear and tear that makes something worth less than before. There are a few key elements when understanding depreciation:

• Time: Depreciation happens over a specified period. For cars, it's usually measured annually.
• Rate: This is the percentage by which the value decreases each year. Here, our car depreciates at a rate of 9% per year.

In the exercise mentioned, this tells us that every year the car's value is 9% less than it was the year before. Typically, depreciation is an expected process for most tangible assets. Understanding how depreciation affects an asset's value is crucial for making informed decisions, whether it's for selling, buying, or calculating financial worth.

Exponential Model

An exponential model is a mathematical equation used to describe exponential growth or decay. In this instance, it helps us represent the depreciation of the car in mathematical terms. The general form for an exponential decay model is: \[ P(t) = P_0 \times (1 + r)^t \]Here’s what all the parts mean:

• \(P(t)\): Represents the value of the item (in this case, the car) after a certain amount of time \(t\).
• \(P_0\): This is the initial value, or the starting point, of what you're measuring.
• \(r\): The rate of change, expressed as a decimal.
• \(t\):stands for the time period over which the change is observed, typically in years for assets like cars.

This formula is powerful because it doesn't just apply to cars; it can be used for anything experiencing consistent exponential growth or decay, like populations or radioactive substances.By setting up an exponential model, we're able to predict the future value of the car at any point in time — as long as the rate of depreciation stays constant.

Initial Value

The initial value, often denoted as \(P_0\), is the starting point of any measurement in an exponential model. For our car example, the initial value is the price of the car when it was first purchased, which is $25,000.Understanding initial value is important because:

• It sets the base level from where all changes are measured. Think of it as the zero point in your calculations.
• It's directly used in our exponential formula. This value is multiplied by the decay factor to determine future values.
• Most planning and calculation hinges on knowing this base value, so accuracy is crucial.

In scenarios where the initial value isn't accurately known, predictions about decay or growth might be off. Hence, confirming the initial value helps ensure that your future calculations are reliable and offer a true reflection of changes over time.