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Chapter 2: Problem 3

The resale value \(V\), in dollars, of a certain car is a function of the number of years \(t\) since the year 2008 . In the year 2008 the resale value is \(\$ 18,000\), and each year thereafter the resale value decreases by \(\$ 1700\). a. What is the resale value in the year 2009 ? b. Find a formula for \(V\) as a function of \(t\). c. Make a graph of \(V\) versus \(t\) covering the first 4 years since the year 2008 . d. Use functional notation to express the resale value in the year 2011 , and then calculate that value.

Short Answer

Expert verified
a. $16,300; b. \(V(t) = 18000 - 1700t\); c. Graph points for \(t = 0, 1, 2, 3\); d. \(V(3) = 12900\), value in 2011 is $12,900.

Step by step solution

01

Understanding the Problem

We have a linear depreciation of a car's resale value beginning in the year 2008. The initial value is $18,000, and it decreases by $1,700 each year. This information will help us address each of the provided subquestions.
02

Calculate the Resale Value in 2009

In 2008, the resale value is $18,000. Since the value decreases by $1,700 each year, in 2009, the resale value will be $18,000 - $1,700. Calculate the result of this subtraction.
03

Create the Formula for Resale Value

Each year the value decreases by $1,700. The general formula for linear depreciation based on the number of years, t, since 2008 is: \[V(t) = 18000 - 1700t\]. This equation reflects the resale value as a function of time in years.
04

Graph the Resale Value Function

To graph the function \(V(t) = 18000 - 1700t\) over the first 4 years (t = 0, 1, 2, 3), calculate the resale value for each year and plot these points: \(V(0)=18000\), \(V(1)=16300\), \(V(2)=14600\), \(V(3)=12900\). Draw a line through these points to represent the change over time.
05

Find Resale Value for 2011

2011 is 3 years after 2008, so substitute \(t = 3\) into the function: \(V(3) = 18000 - 1700 imes 3\). Calculate the value to find the resale value in 2011.

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Key Concepts

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

Resale Value
When considering the sale of a used car, understanding the resale value is crucial. Resale value is the projected market price of the car on the second-hand market, based on its age and depreciation.
  • In 2008, the initial resale value of the car is given as $18,000.
  • Each subsequent year, this value decreases by $1,700 due to depreciation, a common decrease in value over time for cars.
  • To find the resale value for any given year, we subtract the cumulative depreciation from the initial value.
Thus, for 2009, you subtract $1,700 once, yielding a resale value of $16,300. The ongoing depreciation affects the car's selling price unless measures are taken to maintain or enhance its condition.
Linear Function
A linear function is a powerful mathematical tool for modeling relationships between variables. In the context of car resale value, it helps predict prices over time.
The formula derived for this specific linear function representing a car's resale value is:
\[ V(t) = 18000 - 1700t \]
  • Here, \( V(t) \) represents the resale value after \( t \) years.
  • The initial value of $18,000 is the y-intercept, showing the car's value in 2008.
  • The coefficient \( -1700 \) indicates the yearly depreciation rate.
Such a function allows you to conveniently calculate the car's price at any future time \( t \), simply by plugging the desired number of years into the equation. This linear relationship is an indicator that the decrease in value remains constant each year.
Graphing Functions
Graphing functions turn abstract equations into visual insights. By plotting the linear function representing the car's resale value, you can see how its price changes over time.
  • Begin by computing values: for \( t = 0, 1, 2, \text{and } 3\), calculate \( V(t) \) using the equation \( V(t) = 18000 - 1700t \).
  • Plot these points: \( (0, 18000) \), \( (1, 16300) \), \( (2, 14600) \), and \( (3, 12900) \).
  • Connect the dots to form a downward-sloping line, reflecting consistent annually decreasing value.
This graph provides a visual explanation of how each year impacts the car's market value. By examining the slope, you can confirm that the rate of depreciation is constant, reinforcing the linearity of the depreciation.

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Most popular questions from this chapter

The Harvard Step Test was developed \({ }^{2}\) in 1943 as a physical fitness test, and modifications of it remain in use today. The candidate steps up and down on a bench 20 inches high 30 times per minute for 5 minutes. The pulse is counted three times for 30 seconds: at 1 minute, 2 minutes, and 3 minutes after the exercise is completed. If \(P\) is the sum of the three pulse counts, then the physical efficiency index \(E\) is calculated using $$ E=\frac{15,000}{P}. $$ The following table shows how to interpret the results of the test. $$ \begin{array}{|l|l|} \hline \text { Efficiency Index } & \text { Interpretation } \\ \hline \text { Below } 55 & \text { Poor condition } \\ \hline 55 \text { to } 64 & \text { Low average } \\ \hline 65 \text { to } 79 & \text { High average } \\ \hline 80 \text { to } 89 & \text { Good } \\ \hline 90 \& \text { above } & \text { Excellent } \\ \hline \end{array} $$ a. Does the physical efficiency index increase or decrease with increasing values of \(P\) ? Explain in practical terms what this means. b. Express using functional notation the physical efficiency index of someone whose total pulse count is 200 , and then calculate that value. c. What is the physical condition of someone whose total pulse count is 200 ? d. What pulse counts will result in an excellent rating?

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