91Ó°ÊÓ

[Part (e) requires use of the Internet and technology to find a best-fit function.] A "rule of thumb" used by car dealers is that the trade-in value of a car decreases by \(30 \%\) each year. a. Is this decline linear or exponential? b. Construct a function that would express the value of the car as a function of years owned. c. Suppose you purchase a car for \(\$ 15,000 .\) What would its value be after 2 years? d. Explain how many years it would take for the car in part (c) to be worth less than \(\$ 1000\). Explain how you arrived at your answer. e. Internet search: Go to the Internet site for the Kelley Blue Book (www.kbb.com). i. Enter the information about your current car or a car you would like to own. Specify the actual age and mileage of the car. What is the Blue Book value? ii. Keeping everything else the same, assume the car is I year older and increase the mileage by 10,000 . What is the new value? iii. Find a best-fit exponential function to model the value of your car as a function of years owned. What is the annual decay rate? iv. According to this function, what will the value of your car be 5 years from now?

Step by step solution

01

Determine the Type of Decline

To determine if the decline is linear or exponential, observe the pattern: the trade-in value decreases by 30% each year. This indicates an exponential decline because the percentage decrease is consistent relative to the car's current value.

02

Construct the Function

To construct a function that expresses the value of the car over time, use the exponential decay formula: Let the initial value of the car be denoted as \(V_0\) and the number of years as \(t\). The value of the car after t years can be represented as: \( V(t) = V_0 \times (0.70)^t \) since the car retains 70% of its value each year.

03

Calculate the Value After 2 Years

Given the initial car price of \(15,000, use the function to calculate its value after 2 years: \( V(2) = 15000 \times (0.70)^2 \). Perform the calculation: \( V(2) = 15000 \times 0.49 = 7350 \). Therefore, the value of the car after 2 years is \)7,350.

04

Determine Number of Years to Reach Below \(1,000

To find when the car's value drops below \)1,000, set up the inequality: \( 1000 > 15000 \times (0.70)^t \). Solve for t: \( (0.70)^t < \frac{1000}{15000} \). \( 0.70^t < 0.0667 \). Taking the natural logarithm on both sides, solve for t: \( t > \frac{\ln(0.0667)}{\ln(0.70)} \). Use a calculator to find: \( t > \frac{-2.7080502}{-0.356675} \approx 7.6 \). So, it would take approximately 8 years for the car to be worth less than \(1,000.

05

Internet Search - Find Blue Book Values

Visit www.kbb.com, enter the car's details (make, model, year, mileage), and note the current value of the car. Then, adjust the car's age by one year and increase the mileage by 10,000 miles, and record the new value.

06

Find Best-fit Exponential Function

Use the Blue Book values obtained to fit an exponential function of the form: \( V(t) = V_0 \times e^{-kt} \). Calculate the decay rate (k) based on the values and find the best-fit function.

07

Calculate Future Value Using Best-fit Function

With the best-fit exponential function determined, calculate the value of the car 5 years from now by substituting t = 5 into the function and solving for V(5).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions

An exponential function is a mathematical expression in the form of \( f(x) = a \times b^x \), where:

• a is the initial value
• b is the base or growth/decay factor
• x is the exponent

Exponential functions describe situations where quantities grow or decay at a constant percentage rate.
For example, if you start with \(100 and it grows by 5% each year, after one year you would have \)105 (\(100 \times 1.05 = 105\) ).
These functions are crucial for modeling real-world phenomena like population growth, radioactive decay, and financial investments.

Financial Mathematics

Financial mathematics involves using mathematical tools and techniques to solve problems related to finance.
Understanding exponential functions is crucial in financial mathematics, especially for modeling investments, loans, and depreciation.
In the context of car value depreciation, we use exponential decay functions to model how the value of a car decreases over time.
For instance, if a car loses 30% of its value each year, we can express its value after t years with the exponential decay formula:\( V(t) = V_0 \times (0.70)^t \).
This allows us to predict future values based on the consistent percentage decrease.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a proportional rate over time.
In exponential decay, the value diminishes rapidly at first and then at a slower rate.
The general form of an exponential decay function is:\( V(t) = V_0 \times e^{-kt} \),
where:

• V(t) is the value at time t
• V_0 is the initial value
• e is the base of the natural logarithm (approximately 2.718)
• k is the decay constant

Using the example of car value depreciation, we can see that the car, initially worth \(15,000, decreases in value by 30% annually.
The function becomes:\( V(t) = 15000 \times (0.70)^t \),
After 2 years, the value is:\( V(2) = 15000 \times (0.70)^2 = 7350 \).
This means that the car would be worth \)7350 after 2 years.
To find when the car's value drops below \(1000, we solve the inequality:\( 1000 > 15000 \times (0.70)^t \).
Simplifying this with logarithms leads to:\( t > \frac{ \ln(0.0667)}{ \ln(0.70)} \approx 7.6 \).
Therefore, it takes approximately 8 years for the car to be worth less than \)1000.