Problem 92 - Precalculus

Question

In \(2005,\) a small business purchased a copier for \(\$ 4500 .\) By \(2008,\) the value of the copier had decreased to \(\$ 3300 .\) Assuming the depreciation is linear, (a) find the rate-of-change \(m=\frac{\Delta \text { value }}{\Delta \text { time }}\) and discuss its meaning in this context. (b) Find the depreciation equation and (c) use the equation to predict the copier's value in 2012 . (d) If the copier is traded in for a new model when its value is less than \(\$ 700\), how long will the company use this copier?

Step-by-Step Solution

Answer

The rate of change is -400. The depreciation equation is \(y = -400x + 4500\), predicting \$1700 in 2012. The copier is used for 9 years.

Step 1: Calculate the Rate of Change

To find the rate of change, use the formula for slope: \( m = \frac{\text{Change in Value}}{\text{Change in Time}} \). The change in value from 2005 to 2008 is \( 3300 - 4500 = -1200 \) dollars. The change in time is \( 2008 - 2005 = 3 \) years. Thus, the rate of change is \( m = \frac{-1200}{3} = -400 \). This means the copier's value decreases by \$400 per year.

Step 2: Write the Depreciation Equation

Using the equation of a line \( y = mx + b \), where \( y \) is the value, \( m \) is the rate of change, and \( b \) is the initial value of the copier in 2005, substitute \( m = -400 \) and \( b = 4500 \). Thus, the equation is \( y = -400x + 4500 \).

Step 3: Predict the Value in 2012

To find the value in 2012, calculate the number of years from 2005 to 2012, which is \( 2012 - 2005 = 7 \) years. Substitute \( x = 7 \) into the depreciation equation: \( y = -400(7) + 4500 = -2800 + 4500 = 1700 \). Thus, the predicted value in 2012 is \$1700.

Step 4: Determine Use Duration Until Value is Below $700

Set \( y = 700 \) and solve for \( x \) in the equation \( 700 = -400x + 4500 \). Rearrange to find \( 400x = 4500 - 700 = 3800 \), so \( x = \frac{3800}{400} = 9.5 \). The company will use the copier for 9 full years (until halfway through the 9th year it drops below $700). Therefore, by the end of the 9th year (2014), it's ready for trade in.

Key Concepts

Rate of Change

The rate of change gives us insight into how quickly something is increasing or decreasing. In the context of depreciation, it shows the average amount by which the value of an asset decreases each year. In our example, the rate of change (\( m = \frac{\text{Change in Value}}{\text{Change in Time}} \)) is calculated based on the depreciation of a copier over time. Here, the copier's value dropped from \(4500 in \(2005\) to \)3300 in \(2008\). The equation to find this rate is:

• Change in Value: \(3300 - 4500 = -1200\) dollars
• Change in Time: \(2008 - 2005 = 3\) years

So, the rate of change \( m \) is \( \frac{-1200}{3} = -400 \). This means the copier's value decreases by $400 each year. Understanding this concept helps in predicting future values and managing financial expectations.

Depreciation Equation

A depreciation equation is a mathematical representation allowing us to model and predict the decreasing value of an asset over time. It typically uses the structure of a linear equation: \( y = mx + b \), where \( y \) represents the value at any point in time, \( m \) is the rate of change, and \( b \) is the initial value. For our copier example:

• Initial value in 2005
  \( b = 4500 \)
• Rate of change
  \( m = -400 \)

Substituting these into the equation gives us: \( y = -400x + 4500 \). This equation will help in illustrating how the value changes over time and is fundamental to understanding the linear nature of depreciation.

Predicting Value

Predicting the value of depreciating assets is crucial in financial planning. With our depreciation equation, \( y = -400x + 4500 \), we can estimate the value at any future date. For instance, to find out what the copier is worth in \(2012\), calculate the number of years elapsed from \(2005\) which is \(7\) years:

• Substitute \( x = 7 \) into the equation: \( y = -400(7) + 4500 \)
• Follow the calculation:
  \( y = -2800 + 4500 = 1700 \)

Thus, the copier's predicted value in \(2012\) is $1700. This predictive approach is essential for budgeting, tax assessments, and business strategy.

Equipment Lifecycle

The lifecycle of equipment includes phases from purchase to eventual disposal. Understanding when equipment should be phased out or replaced is key to maximizing value and efficiency. In the copier example, we determine when its value falls below \(700. First, set this value into the depreciation equation as \( y \). So:

• \( 700 = -400x + 4500 \)

Rearrange the equation to solve for \( x \):

• \( 400x = 4500 - 700 = 3800 \)
• \( x = \frac{3800}{400} = 9.5 \)

This calculation shows the copier will operate for \(9\) full years before its value drops below \)700. Thus, by the end of \(2014\), it's ready for trade-in. Understanding the lifecycle helps in strategic planning of asset replacement.