Depreciation: Once a piece of equipment is put into service,...

Depreciation: Once a piece of equipment is put into service, its value begins to depreciate. A business purchases some computer equipment for $\$ 18,500 .$ At the end of a 2 -yr period, the value of the equipment has decreased to $\$ 11,500 .$ (a) Use the relation (time since purchase, value) to find a linear equation modeling the value of the equipment. (b) Discuss what the slope and $y$ -intercept indicate in this context. (c) What is the equipment's value after 4 yr? (d) How many years after purchase will the value decrease to $\$ 6000 ?$ (e) Generally, companies will sell used equipment while it still has value and use the funds to purchase new equipment. According to the function, how many years will it take this equipment to depreciate in value to $\$ 1000 ?$

Depreciation: Once a piece of equipment is put into service, its value begins to depreciate. A business purchases some computer equipment for $\$ 18,500 .$ At the end of a 2 -yr period, the value of the equipment has decreased to $\$ 11,500 .$ (a) Use the relation (time since purchase, value) to find a linear equation modeling the value of the equipment.
(b) Discuss what the slope and $y$ -intercept indicate in this context. (c) What is the equipment's value after 4 yr? (d) How many years after purchase will the value decrease to $\$ 6000 ?$ (e) Generally, companies will sell used equipment while it still has value and use the funds to purchase new equipment. According to the function, how many years will it take this equipment to depreciate in value to $\$ 1000 ?$

Key Concepts

Extrapolation and Prediction

Extrapolation in the context of linear depreciation involves using the linear model to predict the asset’s value outside the range of the given data. This is useful for estimating future values, such as how long it will take for the asset to depreciate to a specific value or what its value will be at a particular future time. By solving the linear equation for different values of time or value, decision makers can make educated forecasts and plan for asset replacement or resale.

Slope in Linear Equations

In the context of a linear depreciation model, the slope represents the rate of change in the asset’s value for every unit of time that passes. Essentially, it quantifies how much the asset loses in value per year or per period. A negative slope indicates a decline in value over time, and understanding this rate allows businesses to estimate future loss in value or determine when an asset might reach a certain residual value.

Y-Intercept in Linear Equations

The y-intercept in a linear equation is the point where the graph intersects the y-axis, which in depreciation modeling represents the initial value of the asset at time zero. This starting value is critical as it sets the baseline for all subsequent calculations of value decline over time. It ensures that the model accurately reflects the asset’s value at the moment it is put into service.

Depreciation

Depreciation is the process by which an asset's value decreases over time due to factors such as wear and tear, obsolescence, or market conditions. It’s an important accounting and financial concept used to systematically allocate the cost of a tangible asset over its useful life. This concept is not only essential for reflecting the realistic value of an asset on financial statements but also for making informed economic decisions regarding asset replacement and investment.

Linear Function Modeling

Linear function modeling involves representing a relationship between two variables using a linear equation, commonly expressed as y = mx + b. In the context of depreciation, a linear model assumes that the asset’s value decreases consistently over time. This method is useful for simplifying and predicting the value reduction of an asset, and it allows decision makers to establish clear expectations for the asset’s future worth.

Transcript

00:01 We are told that a piece of equipment was purchased for $18 ,500 and two years after it has been put into use, and now has a value of $11 ,500.

00:11 So as you can see, the value of this piece of equipment decreases as it has been put in use for a longer amount of time.

00:18 We need to come up with an equation that models the value of this equipment over time.

00:23 In order to come up with an equation, we need points and the slope.

00:27 So let's convert these into points.

00:29 Let's say that the x value here is going to be time.

00:33 That is the time since the equipment was purchased.

00:36 So when time is zero, that is when the equipment has just been purchased, its value is $18 ,500.

00:43 So we have the point zero comma, 18 ,500.

00:46 Then two years later, it has a value of $11 ,500, meaning we have the point two comma, 11 ,500.

00:56 Now we have these two points and we can start making an equation.

00:59 First up, we're going to need the slope.

01:01 Slope is given by the slope formula, m equals y2 minus y1, divided by x2 minus x1, which you should have gotten from the book.

01:10 So plugging in some values for this, where x1, y1 is a point, and x2y2 is another point, preferably with larger x, we're going to get the following.

01:21 We'll have 11 ,500 minus 18 ,500.

01:27 That's just subtracting the y's, and then we subtract the xes in the same order.

01:31 2 minus 0.

01:35 All right, this will come down to negative 7 ,000 divided by 2, which one divided out gives negative 3 ,500.

01:44 That is the slope of this equation.

01:47 Now we need to plug this in to point slope form in order to create a full equation.

01:53 Point slope form has the general formula, y1, or sorry, y minus y1, is equal to m times x minus x1, where m is the slope and x1 y1 is some point.

02:05 Okay, let's plug in what we have.

02:08 So we've got y minus 18 ,500 is equal to m negative 3 ,500 x minus 0, because zero is the associated x value with y equals 18 ,500.

02:25 Now let's clean this up a little bit, y minus 18 ,500.

02:30 Is equal to negative 3 ,500 x, and then adding this 18 ,500 to both sides gives y is equal to negative 3 ,500x plus 18 ,500.

02:45 That is the equation modeling this scenario.

02:49 Before we do anything else, we need to talk about what this means.

02:52 Now, many problems you have been doing in this book are just numbers.

02:56 However, this one is based off of real -world situations.

03:00 The slope and the y intercept have real meaning here.

03:02 Let's examine them up close.

03:04 First off, we're going to look at the y intercept.

03:06 This is defined as the point where x is equal to zero.

03:09 So, as you can see up here, the y intercept is going to be 0 ,18 ,500.

03:16 Alternatively, looking at our equation, since this is in what's called slope intercept form, you can see this plus 18 ,500, and whatever number is added on, not multiplied by x, becomes the y intercept.

03:28 So that's our y intercept.

03:29 What does this mean in the real world situation? well, x is time since purchase, and y is price.

03:36 So the y intercept, the point where x equals zero, is the price when no time has passed.

03:41 That is the initial price of this piece of equipment.

03:46 Let me just write that in here...

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