Problem 76 - Introductory and Intermediate Algebra for College Students 4th

Question

A new car worth 24,000 dollars is depreciating in value by 3000 dollars per year. The mathematical model $$y=-3000 x+24,000$$ describes the car's value, \(y,\) in dollars, after \(x\) years. a. Find the \(x\) -intercept. Describe what this means in terms of the car's value. b. Find the \(y\) -intercept. Describe what this means in terms of the car's value. c. Use the intercepts to graph the linear equation. Because \(x\) and \(y\) must be nonnegative (why?), limit your graph to quadrant I and its boundaries. d. Use your graph to estimate the car's value after five years.

Step-by-Step Solution

Answer

From the equation, the x-intercept is 8 years, meaning the car will have no value after 8 years. The y-intercept is $24,000, showing the initial price of the car. Plotting these in a graph, it shows the depreciation of the car's value over time, from which it is estimated the car's value will be $9000 after five years.

Step 1: Find x-intercept

To find the x-intercept, set \(y=0\) in the equation and solve for \(x\). The x-intercept represents the time in years when the car will not have any monetary value.

Step 2: Find y-intercept

To find the y-intercept, set \(x=0\) in the equation and solve for \(y\). The y-intercept represents the initial value of the car before any depreciation, which in this case is when \(x=0\), i.e., in the beginning.

Step 3: Plot Linear Equation

Plot the equation, representing time(in years) on the x-axis and car's value (in dollars) on the y-axis. Assume positive values for both x and y, because they represent time and money respectively, which cannot be negative.

Step 4: Estimate Car's Value After Five Years

With the car's depreciation modeled in the graph, find the point on the graph that corresponds to \(x=5\) (five years from now). This value will be the car's estimate value after five years.

Step 5: Interpret the intercepts

Interpret the x and y intercepts in terms of the car's value which will help in understanding the depreciation of the car over the years

Key Concepts

Depreciation

Depreciation is the gradual decrease in the value of an asset, like a car, over time. This decline in value occurs due to factors such as wear and tear, or obsolescence. In the context of our exercise, depreciation is modeled using the linear equation \( y = -3000x + 24000 \).

• The negative sign in front of 3000 indicates that the value of the car decreases by 3000 dollars each year.
• The initial price of the car is 24,000 dollars, representing its value when purchased new.

Depreciation impacts financial decisions as it determines how much value is lost each year. This information is crucial for understanding the long-term worth of an asset. Calculating depreciation helps in tax deductions and budgeting for future purchases.
In our case, the car will eventually be worthless as its depreciated value reaches zero. This situation is calculated at the x-intercept, which occurs when the entire initial value is lost over the years.

X-Intercept

The x-intercept is where the graph of an equation crosses the x-axis. In our depreciation problem, this value tells us when the car's worth will completely depreciate to zero. To find the x-intercept:

• Set \( y = 0 \) in the equation \( y = -3000x + 24000 \).
• Solve for \( x \) to get \( x = \frac{24000}{3000} = 8 \).

This means that after 8 years, the car will have no resale value at all. At this point, it has depreciated by 24,000 dollars. Understanding the x-intercept helps in predicting how long an asset can hold its value before becoming economically obsolete.

Y-Intercept

The y-intercept is where the line crosses the y-axis. It represents the starting value when the time is zero. In our scenario, the y-intercept is found by setting \( x = 0 \) in the equation \( y = -3000x + 24000 \):

• This simplifies to \( y = 24000 \).

Here, the y-intercept shows that the initial price of the car is 24,000 dollars before any depreciation occurs. Understanding the y-intercept is crucial because it provides a reference point for how much value will be lost over time due to depreciation. By knowing the starting value, we can calculate how quickly the asset depreciates annually.

Graphing Linear Equations

Graphing the linear equation is a visual way to understand how depreciation affects car value over time. Use a graph with:

• The x-axis representing time in years.
• The y-axis showing the car's value in dollars.

By plotting the y-intercept (0, 24000) and x-intercept (8, 0), we draw a straight line. Limit your graph to the first quadrant where both \( x \) and \( y \) are positive, because time can't be negative, and asset value is never negative. This visual tool helps to predict future values and determine how quickly an asset is losing value as time progresses. Graphing provides not only a numerical but a visual understanding, offering insights into financial decision-making and planning.