Question

Straight-Line Depreciation A small business purchases a piece of equipment for \(\$ 875 .\) After 5 years, the equipment will be outdated, having no value. (a) Write a linear equation giving the value \(y\) of the equipment in terms of the time \(x\) in years, \(0 \leq x \leq 5\). (b) Find the value of the equipment when \(x=2\). (c) Estimate (to two-decimal-place accuracy) the time when the value of the equipment is \(\$ 200\).

Short answer

The linear equation giving the value of the equipment in terms of time is \(y = -175x + 875\). The value of the equipment after 2 years is $525. The equipment will have a value of $200 after approximately 3.86 years.

Step-by-step solution

Write the Linear Depreciation Formula

The linear depreciation formula is \(y = mx + b\), where y is the value of the equipment at time x, m is the rate of depreciation, and b is the initial value of the equipment. The equipment loses $875 in value over 5 years, so the rate of depreciation \(m\) is \(-875 / 5 = -175\). The initial value \(b\) is the initial purchase price which is $875. So the linear equation that gives the value of the equipment at any time \(x\) is \(y = -175x + 875\).

Find the Value of the Equipment at \(x = 2\)

Substitute \(x = 2\) into the linear equation: \(y = -175*2 + 875 = 525\). So the value of the equipment after 2 years is $525.

Estimate the Time When Equipment Value is $200

Substitute \(y = 200\) into the equation and solve for \(x\). This gets \[200 = -175x + 875\]. After rearranging terms: \[-175x = 200 - 875 = -675\]. Finally, after simplifying, \(x = -675/-175 = 3.86\). So approximately at the end of year 3 (or start of year 4), the value of the equipment will fall to $200.

Key concepts

Linear Depreciation Formula

Understanding the linear depreciation formula is essential when evaluating how an asset loses value over time in a fixed manner. This technique spreads the cost of an asset evenly across its useful life, which is highly relevant in fields like accounting and finance.

Here's a simplified breakdown of the linear depreciation formula: \[\begin{equation}y = mx + b\text{,}\text{where}\begin{itemize}\item \( y \) is the remaining value of the asset at time \( x \),\item \( m \) represents the fixed depreciation rate, which is constant every year, and\item \( b \) is the initial purchase price of the asset, also known as the book value at the start of its useful life.\end{itemize}\end{equation}\]In practice, to write this equation for an actual asset, one would need to know the initial cost and the expected duration of its utility period, which is when the asset is considered to have zero salvage value.

Depreciation Rate Calculation

Calculating the depreciation rate for an asset is a critical step in understanding how it will lose value year by year. The rate of depreciation is found with the following formula:\[\begin{equation}m = \frac{(b - s)}{n}\text{,}\end{equation}\]where \( m \) is the depreciation rate, \( b \) is the asset's initial purchase price, \( s \) represents the salvage value (the expected value at the end of the useful life, which often is zero), and \( n \) is the number of years in the asset's useful life. This formulation ensures a straight-line consistent reduction in value over time.

For our specific example, considering the equipment bought at \$875 with no salvage value after 5 years, the depreciation rate calculates to: \[\begin{equation}m = \frac{875 - 0}{5} = 175\text{,}\end{equation}\]showing that the asset devalues by \$175 each year. Here, the negative sign in the formula accounts for the downward trajectory of the asset's value.

Value Estimation Over Time

Estimating the future value of an asset over time involves plugging in different values for time \( x \) into our straight-line depreciation formula. It allows businesses to project when an asset will decrease to a specific value or to determine its present value after a certain number of years.

For the given exercise, we can find the value of the equipment at any point in time by using \( y = -175x + 875 \). For instance, two years down the line (when \( x = 2 \)), the calculation is straightforward:\[\begin{equation}y = -175(2) + 875 = 525\text{,}\end{equation}\]indicating a remaining value of \$525. Alternatively, if we want to find out when the asset's value will drop to \$200, we can rearrange the formula to solve for time:\[\begin{equation}200 = -175x + 875\end{equation}\]\[\begin{equation}x \approx 3.86\text{,}\end{equation}\]which tells us that in approximately 3.86 years (or towards the end of the third year, start of the fourth year), the equipment's value will be close to \$200. These calculations empower businesses to make informed decisions on asset management and budgeting for replacements or upgrades.