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Chapter 2: Problem 81

Find a linear equation in slope-intercept form that models the given description. Describe what each variable in your model represents. Then use the model to make a prediction. A computer that was purchased for 4000 dollar is depreciating at a rate of 950 dollar per year.

Short Answer

Expert verified
The linear equation that represents the depreciation of the computer's value is \(y = -950x + 4000\) where \(y\) is the value of the computer at year \(x\). This can be used to predict the value of the computer in the future. For example, after 2 years, the computer is estimated to be worth $2100.

Step by step solution

01

Identify the Slope and Y-Intercept

The initial cost of the computer, $4000, is the starting point, so it is the y-intercept. The computer is depreciating at a rate of $950 per year, which will be the slope of the line. Because the value of the computer is decreasing over time, the slope will be negative.
02

Write the Equation

The linear equation in slope-intercept form is given by \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. Substituting \(m = -950\) and \(b = 4000\), the equation becomes \(y = -950x + 4000\). \(y\) represents the value of the computer at any given year \(x\)
03

Make a Prediction

This model can be used to predict the value of the computer in the future. For example, to predict the value after 2 years, substitute \(x = 2\) into the equation to get: \(y = -950*2 + 4000 = 2100\). After 2 years, the computer is estimated to be worth $2100. This model assumes that the rate of depreciation is constant over the years.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a simple way to express linear equations. It is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This form is incredibly useful because it allows us to immediately see how one variable directly affects another.
In the context of the exercise, the equation \(y = -950x + 4000\) is in slope-intercept form. Here, \(y\) represents the computer's value over time, and \(x\) represents the number of years since purchase.
This form is especially handy for making predictions or understanding the relationship between the two variables at a glance.
Depreciation
Depreciation is the process by which an asset loses value over time. In this exercise, we observe the computer losing value at a steady rate of $950 per year, which is a common scenario with electronics as they become obsolete.
It's important to understand depreciation because it impacts both personal and business financial decisions immensely.
Knowing an asset's depreciation helps in planning its replacement and understanding its true expense over time.
  • This depreciation is modeled as the negative slope in our linear equation, indicating a decline in value every year.
  • The concept assumes a constant rate of depreciation for simplicity.
Slope of a Line
The slope of a line describes how steep the line is. It indicates the rate of change between two variables. In linear equations, it's represented by \(m\) in the formula \(y = mx + b\).
In our exercise, the slope is \(-950\). This negative slope reflects that the computer's value decreases by $950 each year.
Slope can be thought of as "rise over run," where the rise is the change in \(y\) and the run is the change in \(x\).
  • A positive slope indicates an increase, while a negative slope indicates a decrease.
  • A zero slope means there's no change over time, implying a horizontal line.
Y-Intercept
The y-intercept of a linear equation is the point where the line crosses the y-axis. It is represented by \(b\) in the slope-intercept form \(y = mx + b\). This value is the \(y\)-coordinate when \(x\) is zero.
In the exercise, the y-intercept is 4000. This means before any time has passed, the computer is initially worth $4000.
The y-intercept serves as a starting value, giving the initial condition before any changes modeled by the slope occur.
  • This intercept helps in understanding the initial condition of the scenario being analyzed.
  • It's a crucial component when constructing graphs of linear relationships.

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Most popular questions from this chapter

Find a linear equation in slope-intercept form that models the given description. Describe what each variable in your model represents. Then use the model to make a prediction. In \(1995,\) the average temperature of Earth was \(57.7^{\circ} \mathrm{F}\) and has increased at a rate of \(0.01^{\circ} \mathrm{F}\) per year since then.

Which one of the following is true? a. If \(f(x)=|x|\) and \(g(x)=|x+3|+3,\) then the graph of \(g\) is a translation of three units to the right and three units upward of the graph of \(f\) b. If \(f(x)=-\sqrt{x}\) and \(g(x)=\sqrt{-x},\) then \(f\) and \(g\) have identical graphs. c. If \(f(x)=x^{2}\) and \(g(x)=5\left(x^{2}-2\right),\) then the graph of \(g\) can be obtained from the graph of \(f\) by stretching \(f\) five units followed by a downward shift of two units. d. If \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4,\) then the graph of \(g\) can be obtained from the graph of \(f\) by moving \(f\) three units to the right, reflecting in the \(x\) -axis, and then moving the resulting graph down four units.

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)=(x-2)^{3} $$

The graph shows the amount of money, in billions of dollars, of new student loans from 1993 through 2000 . (graph can't copy) The data shown can be modeled by the function \(f(x)=6.75 \sqrt{x}+12,\) where \(f(x)\) is the amount, in billion of dollars, of new student loans \(x\) years after 1993 . a. Describe how the graph of \(f\) can be obtained using transformations of the square root function \(f(x)=\sqrt{x} .\) Then sketch the graph of \(f\) over the interval \(0 \leq x \leq 9 .\) If applicable, use a graphing utility to verify your hand-drawn graph. b. According to the model, how much was loaned in \(2000 ?\) Round to the nearest tenth of a billion. How well does the model describe the actual data? c. Use the model to find the average rate of change, in billions of dollars per year, between 1993 and 1995 Round to the nearest tenth. d. Use the model to find the average rate of change, in billions of dollars per year, between 1998 and 2000 . Round to the nearest tenth. How does this compare with you answer in part (c)? How is this difference shown by your graph? e. Rewrite the function so that it represents the amount, \(f(x),\) in billions of dollars, of new student loans \(x\) years after 1995

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$ r(x)=(x-3)^{3}+2 $$
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