Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

\((p .324)\) In early \(2014,\) a new sedan had a manufacturer's suggested retail price of \(\$ 31,340 .\) After \(t\) years, the sedan's value is given by \(V(t)=31,340\left(\frac{4}{5}\right)^{t}\) (a) Use a graphing utility to graph the function. (b) Use the graphing utility to create a table of values that shows the value \(V\) for \(t=1\) to \(t=10\) years. (c) According to the model, when will the sedan have no value?

Step-by-Step Solution

Answer

For part (c), the exact time when the car becomes valueless can't be calculated with the given model because, mathematically, the value of the car will never become exactly zero. But it will continually get closer to zero as time progresses.

Step 1: Graph the Function

To graph the function \(V(t)=31,340\left(\frac{4}{5}\right)^{t}\), input the function into any graphing utility. The x-axis should represent the number of years \(t\), and the y-axis should represent the value of the car \(V\).

Step 2: Create a Table of Values

Using the graphing utility, generate a table of values for the function. It should start at \(t=1\) and end at \(t=10\). The table will show the car value \(V\) depending on the time \(t\). We can see that as \(t\) increases, the value \(V\) of the car decreases.

Step 3: Find the Time When the Car Has No Value

To find out when the value of the car will be zero, we need to solve \(V(t)=31,340\left(\frac{4}{5}\right)^{t} = 0\) for \(t\). However, for a continuously compounded decay function, the time \(t\) when the value \(V(t)\) becomes exactly zero is mathematically infinite. In other words, the car's value will never become exactly zero. It will only get smaller and smaller.

Key Concepts

Exponential Decay Function

An exponential decay function is a type of mathematical expression that models how a quantity decreases at a rate proportional to its current value. In the context of car depreciation, the formula given in the exercise, \(V(t)=31,340\left(\frac{4}{5}\right)^{t}\), is a classic example of an exponential decay function. Here, \(V(t)\) represents the value of the car after \(t\) years, and the \(\frac{4}{5}\) component signifies that the car retains 80% of its value per year.

This concept is pivotal in understanding how quickly the value of an asset deteriorates over time. It’s important to note that the function never actually reaches zero but gets infinitesimally close, thus reflecting the idea that a car will always retain some level of value, albeit very small after a long time period.

As the number of years increases, the car's value is multiplied by \(\left(\frac{4}{5}\right)\) repeatedly, which is why we see a rapid shrinkage in value initially, which tapers off as the value gets lower. This is a fundamental concept when studying depreciation and forms the basis of how many assets' value diminishes over time.

Graphing Functions

Graphing functions is a visual means of understanding mathematical relationships. With a graph, it’s easier to comprehend how the value of an object like a car decreases over time. When graphing the exponential decay function mentioned above, you'll typically have the \(t\) value, or time, on the x-axis, while the y-axis will show the car's value, \(V(t)\).

A significant point while graphing is to observe the shape of the curve. For an exponential decay function, the graph will start high when \(t=0\) and then curve downwards more steeply at first before flattening out as the value decreases. Graphing thus provides a clear and instantaneous understanding of how variables interact, in this case, how time affects the car's depreciation.

Value Depreciation

Value depreciation is the reduction in the monetary worth of an asset over time. It's a significant concept in economics and finance, particularly relevant to items like cars, which tend to lose value rapidly. The depreciation rate can vary based on factors such as make, model, market conditions, and usage.

Understanding how value depreciation works can help individuals make better financial decisions, like calculating the return on an investment or understanding when is the most cost-effective time to replace an asset. In our example, the depreciation model is 20% annually, which means that every year the car loses 20% of its value compared to the previous year. This rate of depreciation informs both the current value of the car and its projected future value, critical for both buyers and sellers in the automotive market.

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-world situations and predict future scenarios. It’s a tool used extensively in various fields such as science, engineering, and economics. In our exercise, the exponential decay function is a mathematical model of how the car's value depreciates over time.

Good models are pivotal for accurate predictions and decision-making. They are not perfect representations of reality, but they provide a simplified version that is easier to manage and understand. Mathematical modeling with decay functions also helps us make sense of the world by providing structured ways to estimate how a particular entity, like a car's value, will change.