Problem 92 - Beginning and Intermediate Algebra

Question

To consult with an attorney costs \(\$ 35\) for every \(10 \mathrm{min}\) or fraction of this time. Let \(\mathcal{C}(t)\) represent the cost of meeting an attorney, and let \(t\) represent the length of the meeting, in minutes. Graph \(C(t)\) for meeting with the attorney for up to (and including) 1 hr.

Step-by-Step Solution

Answer

The cost function for consulting with an attorney for up to 1 hour (60 minutes) is: \[ C(t) = \begin{cases} 35 & \text{if } 0 < t \leq 10 \\ 70 & \text{if } 10 < t \leq 20 \\ 105 & \text{if } 20 < t \leq 30 \\ 140 & \text{if } 30 < t \leq 40 \\ 175 & \text{if } 40 < t \leq 50 \\ 210 & \text{if } 50 < t \leq 60 \\ \end{cases} \] To graph C(t), plot horizontal line segments for each time interval on the coordinate plane with the x-axis representing time (in minutes) and the y-axis representing the cost (in dollars). Each interval should have an open circle at the starting point and a closed circle at the endpoint.

Step 1: Determine the cost function

Since the cost is \(35 for every 10 minutes or fraction of this time, divide 10 minutes intervals and add \)35 for each interval or fraction. The cost function, C(t), can be represented as: \[ C(t) = \begin{cases} 35 & \text{if } 0 < t \leq 10 \\ 70 & \text{if } 10 < t \leq 20 \\ 105 & \text{if } 20 < t \leq 30 \\ ... \\ 35n & \text{if } 10(n-1) < t \leq 10n \\ ... \\ \end{cases} \]

Step 2: Determine the cost function for up to 1 hour (60 minutes)

Since we are asked to graph the cost function for up to and including 1 hour (60 minutes), we only need to consider the intervals within this time frame, i.e., 6 intervals. \[ C(t) = \begin{cases} 35 & \text{if } 0 < t \leq 10 \\ 70 & \text{if } 10 < t \leq 20 \\ 105 & \text{if } 20 < t \leq 30 \\ 140 & \text{if } 30 < t \leq 40 \\ 175 & \text{if } 40 < t \leq 50 \\ 210 & \text{if } 50 < t \leq 60 \\ \end{cases} \]

Step 3: Graph the cost function

Plot the cost function C(t) on the coordinate plane with the x-axis representing time (in minutes) and the y-axis representing the cost (in dollars). Remember, since this is a step function, each interval should be represented as a horizontal line segment with an open circle at the starting point of the time interval and a closed circle at the endpoint. The graph of C(t) will have the following line segments: 1. \((0, 35]\): A horizontal line segment from x = 0 (open circle) to x = 10 (closed circle) at the height of y = 35 2. \((10, 70]\): A horizontal line segment from x = 10 (open circle) to x = 20 (closed circle) at the height of y = 70 3. \((20, 105]\): A horizontal line segment from x = 20 (open circle) to x = 30 (closed circle) at the height of y = 105 4. \((30, 140]\): A horizontal line segment from x = 30 (open circle) to x = 40 (closed circle) at the height of y = 140 5. \((40, 175]\): A horizontal line segment from x = 40 (open circle) to x = 50 (closed circle) at the height of y = 175 6. \((50, 210]\): A horizontal line segment from x = 50 (open circle) to x = 60 (closed circle) at the height of y = 210 Upon completing these steps, the graph will visually represent the cost function C(t) for meeting lengths up to and including 1 hour (60 minutes).

Key Concepts

Step Functions

A step function is a type of piecewise function that is characterized by a series of flat, horizontal line segments. These segments reflect specific intervals where the function maintains a constant value.
It looks like a series of "steps" on a graph.
In the context of our exercise, the attorney's payment schedule acts as a perfect example of a step function.

• Each step corresponds to a specific 10-minute interval.
• The height of each step represents the cost for that interval.

One unique feature of step functions is how they handle endpoints:- An open circle signifies a value that is **not included** in the interval.- A closed circle indicates that a value is **included** in the interval.For example, if the cost of the meeting is \(35\) dollars for any time **up to** and including 10 minutes, the step function will have a closed circle at 10 minutes and an open circle at 0. This representation simplifies the understanding of costs that change in non-continuous increments, like our attorney's fees.

Graphing Functions

Graphing functions like step functions involves plotting points on a coordinate plane and connecting them according to the given rules of the function.
The x-axis typically represents the independent variable—in our case, time in minutes—while the y-axis represents the dependent variable, which is the cost in dollars. When graphing the cost function from our exercise:

• Each horizontal line represents the cost for each 10-minute interval.
• The x-axis denotes time, divided into intervals of 10 minutes.
• The y-axis represents cost, incrementing by \(35\) dollars with each interval.

Key points to remember when graphing step functions include:- **Open circles** at the start of each horizontal segment imply that the function value is not calculated at that point.- **Closed circles** at the end of each horizontal segment imply that the function value is applicable.Visualizing these functions on a graph helps clarify how the function behaves over different intervals and demonstrates why these are called step functions, thanks to their characteristic appearance.

Mathematical Modeling

Mathematical modeling is the process of using mathematical language and tools to represent real-world situations so they can be analyzed and understood.
In the case of our exercise, we use a step function to model the cost of consulting with an attorney. This approach offers several benefits:

• **Simplification:** The function simplifies a complex situation where costs depend on discrete time intervals.
• **Visualization:** By graphing the step function, we can quickly comprehend how the cost changes over time.
• **Predictability:** The model predicts costs without needing to conduct lengthy calculations for each time variation.

Mathematical models depend on accurate assumptions and clearly defined parameters to be effective: - They should capture the essential elements of the situation being studied. - They should be flexible enough to be adjusted if conditions change (for example, if the attorney changes her fees). In our scenario, understanding the step function as a mathematical model helps to forecast potential meeting costs. This model can save clients from misunderstanding fees and ensures transparency in financial planning.