Problem 68 - Algebra and Trigonometry

Question

A small business buys a computer for\(\$ 4000\) . After 4 years the value of the computer is expected to be \(\$ 200\) . For accounting purposes, the business uses linear depreciation to assess the value of the computer at a given time. This means that if \(V\) is the value of the computer at time \(t\) , then a linear equation is used to relate \(V\) and \(t\). (a) Find a linear equation that relates \(V\) and \(t\) (b) Sketch a graph of this linear equation. (c) What do the slope and \(V\) -intercept of the graph represent? (d) Find the depreciated value of the computer 3 years from the date of purchase.

Step-by-Step Solution

Answer

(a) \( V = -950t + 4000 \); (c) \(-950\) is the yearly loss, \(4000\) is the initial value; (d) \$1150 after 3 years.

Step 1: Identify the initial and final values

The computer is purchased for \\(4000, so initially, at time \(t = 0\), the value \(V\) is \\)4000. After 4 years, at \(t = 4\), the value of the computer is \$200.

Step 2: Calculate the slope of the depreciation line

The slope \(m\) of a linear equation \(V = mt + c\) is calculated as the change in value over the change in time. Here, \(m = \frac{200 - 4000}{4 - 0} = \frac{-3800}{4} = -950\). This means the computer loses \$950 in value each year.

Step 3: Establish the linear equation

Using the point-slope form of a line \(V = mt + c\) and knowing \(m = -950\) and the value at \(t = 0\) is \$4000, the equation of the line is: \( V = -950t + 4000 \).

Step 4: Express the relation between V and t

The linear equation that relates \(V\) and \(t\) is \( V = -950t + 4000 \). This equation represents the linear depreciation of the computer's value over time.

Step 5: Interpret the slope and V-intercept

The slope \(-950\) represents a loss of \$950 per year. The \(V\)-intercept \(4000\) is the initial value of the computer at \(t = 0\).

Step 6: Calculate the value after 3 years

Plug \(t = 3\) into the equation \( V = -950t + 4000 \): \( V = -950(3) + 4000 = -2850 + 4000 = 1150 \). The value after 3 years is \$1150.

Key Concepts

Linear Equations

Linear equations describe relationships between two variables using a straight line. They are vital in representing many real-world phenomena, including the depreciation of assets like computers. In a linear equation, the general form is given by \( y = mx + c \), where:

• \( y \) represents the dependent variable,
• \( x \) represents the independent variable,
• \( m \) is the slope of the line, indicating how much \( y \) changes for a unit change in \( x \),
• \( c \) is the y-intercept, showing the value of \( y \) when \( x = 0 \).

This simple and powerful mathematical tool helps predict future values and understand trends over time. It's especially useful in finance for modeling depreciation, where asset values decrease predictably over the years.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to quickly understand the relationship between two variables. The equation \( y = mx + c \) helps identify the key features of a linear relationship with ease. Let's break it down:
- **Slope (\( m \))**: This number tells us how steep our line is, and in the context of depreciation, it shows how much an asset's value falls each year.- **Y-intercept (\( c \))**: This is where the line crosses the y-axis, representing the starting value of the asset.

In our exercise, the equation \( V = -950t + 4000 \) captures the depreciation of a computer's value:

• \( V \) (value of the computer) is decreasing by \\(950 per year (slope).
• The initial value is \\)4000 (y-intercept) when \( t = 0 \).

This straightforward form makes it easy to visualize and calculate changes over time.

Depreciation Value Calculation

Calculating depreciation allows businesses to estimate the declining value of assets like computers over time. The calculation uses the linear equation \( V = -950t + 4000 \) derived from the slope-intercept form. Here's how you can find the depreciation value:

• Identify the initial value \( V \) when \( t = 0 \), which is \\(4000 in this case.
• Understand that the slope (-950) shows a yearly decrease in value by \\)950.
• To find the value at any time \( t \), substitute \( t \) into the equation.

For example, after 3 years, to calculate the computer's value: substitute \( t = 3 \) into the equation:
\[ V = -950 \times 3 + 4000 = 1150 \].
So, after 3 years, the computer is valued at \$1150. This method equips businesses with a systematic approach to track and plan for asset depreciation.