Problem 107 - Calculus Volume 1

Question

ITI A car was purchased for \(\$ 26,000\) . The value of the car depreciates by \(\$ 1500\) per year. a. Find a linear function that models the value \(V\) of the car after \(t\) years. b. Find and interpret \(V(4)\)

Step-by-Step Solution

Answer

The formula is \( V(t) = -1500t + 26000 \) and \( V(4) = 20000 \). After 4 years, the car is worth \$20,000.

Step 1: Understanding the Linear Function Formula

A linear function can be written in the form \( V(t) = mt + b \), where \( V(t) \) represents the value of the car after \( t \) years, \( m \) is the slope representing the rate of depreciation, and \( b \) is the initial value or purchase price of the car.

Step 2: Identifying Key Components

The initial value of the car, \( b \), is \\(26,000, and the car depreciates by \\)1,500 per year. This depreciation rate per year is the slope, \( m = -1500 \), since it's a decrease in value.

Step 3: Formulating the Linear Function

Substitute \( m \) and \( b \) into the linear function formula: \[ V(t) = -1500t + 26000 \]. This formula models how the value of the car decreases over time.

Step 4: Evaluating the Function at 4 Years

To find \( V(4) \), which is the car's value after 4 years, substitute \( t = 4 \) into the linear function: \[ V(4) = -1500(4) + 26000 \].

Step 5: Calculating V(4)

Perform the calculation: \(-1500 \times 4 = -6000\). Then, add this to the initial value: \( 26000 - 6000 = 20000 \). Thus, \( V(4) = 20000 \).

Step 6: Interpreting V(4)

The value \( V(4) = 20000 \) means that after 4 years, the value of the car is expected to be \$20,000.

Key Concepts

Depreciation

Depreciation is a term used to describe the reduction in the value of an asset over time. For a car, this process usually occurs because of wear and tear, usage, and eventually, obsolescence as newer models become available. In our exercise, the car initially worth \( \\(26,000 \) depreciates at a rate of \( \\)1,500 \) each year. Thus, depreciation is a key factor in understanding how the car's value decreases annually.

• Depreciation is often expressed as a negative value because it represents a loss in worth.
• This decrease is considered linear if the value reduces by a consistent amount each year.
• Knowing depreciation helps owners anticipate the future value of their asset over time.

The importance of understanding depreciation is that it allows us to create a realistic financial plan. For instance, with this car, we can predict that its worth decreases steadily every year, which aids in decision-making about future selling or trading options.

Slope-Intercept Form

The slope-intercept form is a mathematical equation of a line. A linear function is typically expressed in this format: \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating how the value changes with each unit increase in \( x \).
• \( b \) is the y-intercept, representing the initial starting value before any changes.

In the context of our exercise, we use the function for modeling the car's depreciation: \( V(t) = -1500t + 26000 \).

• The slope, \( m = -1500 \), reflects the yearly decrease in the car's value.
• The y-intercept, \( b = 26000 \), stands for the car's initial purchase price.

Utilizing the slope-intercept form helps visualize and calculate the change in the car's value over time. It's a practical tool for predicting these changes consistently, as long as the depreciation rate remains constant.

Function Evaluation

Function evaluation involves finding the output of a function for a specific input. In our car depreciation example, evaluating the function \( V(t) = -1500t + 26000 \) at \( t = 4 \) years gives us a way to determine the car's projected value at that point in time.

• To evaluate, substitute the given time (in years) into the linear function where \( t \).
• Then calculate the resulting expression to find the value after that duration.

For \( V(4) \), the calculation steps are as follows: - Replace \( t \) with 4: \( V(4) = -1500(4) + 26000 \) - Compute \( -1500 \times 4 = -6000 \) - Add \( 26000 - 6000 = 20000 \) Thus, the evaluated function, \( V(4) = 20000 \), tells us that after 4 years, the vehicle's estimated value will be \( \$20,000 \). This straightforward process assists in making informed predictions based on the current depreciation model, ensuring accurate planning for future financial aspirations.