Problem 44 - Elementary and Intermediate Algebra

Question

Salvage Value. \(\quad\) A small business purchased a computer for S5,000. The value (in dollars) of the computer, \(t\) years after its purchase, is given by the exponential function \(v(t)=5,000(0.75)^{t}\) a. Graph the function. Scale the \(t\) -axis from 0 to 10 in units of 2\. Scale the \(v(t)\) -axis from 0 to \(6,000\) in units of \(1,000\). b. Use the function to determine the value of the computer 12 years after it is purchased.

Step-by-Step Solution

Answer

The computer's value 12 years after purchase is approximately $158.38.

Step 1: Understand the Function

The given function is an exponential decay function: \( v(t) = 5,000(0.75)^t \). Here, \( v(t) \) represents the value of the computer \( t \) years after purchase, where the initial value is \$5,000 and decreases by 25% each year (since \(0.75 = 1 - 0.25\)).

Step 2: Create a Graph

To graph \( v(t) = 5,000(0.75)^t \), plot the time \( t \) on the horizontal axis, ranging from 0 to 10 in intervals of 2. Plot the value \( v(t) \) on the vertical axis, ranging from 0 to 6,000 in intervals of 1,000. Plot points for each value of \( t \) (e.g., \( t = 0, 2, 4, \, \text{etc.} \)), and draw a smooth curve that decreases from the initial value.

Step 3: Calculate the Value After 12 Years

To find the value of the computer 12 years after purchase, substitute \( t = 12 \) into the function: \( v(12) = 5,000(0.75)^{12} \). Calculate \( (0.75)^{12} \) and multiply by 5,000 to get \[ v(12) = 5,000 imes 0.031676352 \approx 158.38. \]

Step 4: Interpret the Result

The computed value \( v(12) \approx 158.38 \) means that after 12 years, the computer's value has depreciated to approximately \$158.38.

Key Concepts

Graphing Exponential Functions

Graphing an exponential function like \( v(t) = 5,000(0.75)^t \) helps visualize how the value changes over time. To create an accurate graph, it's essential to clearly define both axes. The horizontal axis, representing time \( t \), is scaled from 0 to 10 in intervals of 2 years. The vertical axis, representing the value \( v(t) \), extends from 0 to 6,000 in increments of 1,000 dollars.
To plot the graph, calculate \( v(t) \) for each interval of \( t \). Here’s how:

• At \( t=0 \), the value is \( 5,000 \) dollars.
• At \( t=2 \), calculate \( 5,000(0.75)^2 \).
• Follow similar calculations for \( t=4, 6, 8, 10 \).

Once you have these points, connect them with a smooth curve that starts at \( 5,000 \) and gradually descends, showing the depreciation in value over time.

Depreciation in Value

Depreciation refers to the decrease in value of an asset over time. This concept is crucial in understanding how items like electronics lose worth each year. In the given exercise, the computer’s value decreases by 25% annually, modeled by the function \( v(t) = 5,000(0.75)^t \).
Here's how depreciation works in this context:

• The multipliers \( (0.75) \) indicate a 25% reduction every year.
• Initially, at \( t=0 \), the value is the full amount, \( 5,000 \).
• After one year, the value is \( 75\% \) of \( 5,000 \), or \( 3,750 \).
• As time passes, this pattern continues, exponentially reducing the computer's worth.

Understanding this exponential decline is vital, not just for predicting future values but also for financial planning and accounting.

Algebraic Functions

Algebraic functions, like the exponential decay function in this exercise, play a critical role in modeling real-world scenarios. They use mathematical equations to represent relationships between different quantities.
For the function \( v(t) = 5,000(0.75)^t \), the components are:

• Initial Value: \( 5,000 \) dollars is the starting point of the function, representing the initial cost of the computer.
• Decay Factor: \( 0.75 \) signifies the percentage of value retained after each year, reflecting a 25% depreciation rate.
• Variable \( t \): Represents time in years, affecting the power to which 0.75 is raised. As \( t \) increases, the value of \( v(t) \) reduces.

Using algebraic functions like this allows us to answer questions about how an asset's worth changes over time. We can easily compute future values, analyze trends, and even predict when an item becomes financially unviable.