Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

You buy a personal watercraft for \(\$ 8250 . The depreciated value \)y\( after \)t\( years is y=8250-689 t\) for \(0 \leq t \leq 10 (a) Use the constraints of the model and a graphing utility to graph the equation using an appropriate viewing window. (b) Use the zoom and trace features of the graphing utility to determine the value of \)t\( when \)y=5545.25 .\( Verify your answer algebraically. (c) Use the value feature of the graphing utility to determine the value of \)y\( when \)t=5.5 .$ Verify your answer algebraically.

Step-by-Step Solution

Answer

By graphing the function, you will get a visualization of how the personal watercraft's value depreciates over time. Following the steps and computations, (b) the time when \(y=5545.25\) is approximately \(3.92\) years. And (c) the value of \(y\) when \(t=5.5\) is approximately \(4560.5\).

Step 1: Graph the equation

Use a graphing utility to plot the linear function \(y = 8250 - 689t\) with \(0 \leq t \leq 10\). This will give an overall view of the function.

Step 2: Find the value of t

When \(y = 5545.25\), determine the value of \(t\). This can be investigated using the 'trace' feature of the graphing utility. To affirm the result algebraically, however, solve \(8250 - 689t = 5545.25\) for \(t\).

Step 3: Find the value of y

Determine the value of \(y\) when \(t = 5.5\). Using a graphing tool, this can be achieved by the 'value' function. To validate your answer algebraically, substitute \(t = 5.5\) into the equation, yielding \(y = 8250 - 689 \times 5.5\).

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that helps students visualize mathematical functions through graphical representations. In the context of linear depreciation, a graphing utility can be used to plot the line representing the depreciation of a personal watercraft's value over time. The linear equation given is \( y = 8250 - 689t \), where \( y \) represents the value of the watercraft, and \( t \) represents time in years.

When using a graphing utility, it's important to set the correct viewing window to capture all relevant details. For this equation, the domain \( 0 \leq t \leq 10 \) is specified, so adjust the graph accordingly to ensure the full range is visible. The graph will display as a downward-sloping line, as the value decreases over time.

To gain better insights, you can use functions such as 'zoom' and 'trace'. The zoom feature allows you to look at smaller regions of the graph more closely, while the trace feature lets you move along the line to find specific values of \( t \) or \( y \). By interacting with the graph, you'll better understand how depreciation affects the watercraft's value.

Algebraic Verification

Algebraic verification involves checking results mathematically to ensure accuracy. After using a graphing utility to find values of \( t \) and \( y \), confirm these results algebraically.

For instance, to find \( t \) when \( y = 5545.25 \), solve the equation \( 8250 - 689t = 5545.25 \). Simplify this to find \( t \) by performing the necessary algebraic manipulations.

• Subtract 5545.25 from both sides: \( 2704.75 = 689t \)
• Divide both sides by 689 to solve for \( t \): \( t = \frac{2704.75}{689} \)

Similarly, to verify \( y \) when \( t = 5.5 \), substitute 5.5 into the equation \( y = 8250 - 689 \times 5.5 \). This gives the value of \( y \) directly. Both methods ensure that the results obtained using the graphing utility are correct and consistent. Algebraic verification supports a deeper understanding as it illustrates the mathematical processes behind the graph.

Graphing Linear Functions

Graphing linear functions is essential to understand the relationship between variables, especially in real-world applications like depreciation. Linear functions have the form \( y = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept. In our scenario, the function \( y = 8250 - 689t \) depicts how the item's value decreases linearly over time.

The slope \( -689 \) shows the rate of depreciation per year. It's important to recognize that a negative slope indicates a decline in value. The y-intercept (8250) represents the initial value of the watercraft when \( t = 0 \).

When graphing, it's crucial to mark important points, such as where \( y \) equals specific values, or when it crosses a reference point. Using the graph and understanding the slope provides insight into how steeply the value decreases.

• The line helps predict future values easily.
• Recognizing points where the depreciation might affect decisions is quicker with a graph.

Graphing linear functions not only supports practical applications but also deepens your understanding of how mathematical equations model real-life scenarios.