Problem Set 3

A physician owning a private clinic charges professional fee based on the length of time of consultation as shown on the table below. Represent the physician's professional fee as a function of time of consultation.

Time of Conzultation

Professional Fee

First 5 minutes

500.00

Over 8 up to 15 minutes

850.00

In excess of 15 minutes

Plus 50.00 per minute

To represent the physician's professional fee as a function of the time of consultation, we can break down the problem into different cases based on the given time intervals.

Let's define the time of consultation as \( t \) (in minutes) and the professional fee as \( f(t) \) (in currency units).

### Case 1: \( 0 < t \leq 5 \) (First 5 minutes)
- The professional fee is constant at 500.00.

\[ f(t) = 500.00 \quad \text{for } 0 < t \leq 5 \]

### Case 2: \( 5 < t \leq 15 \) (Over 8 up to 15 minutes)
- The professional fee increases to 850.00.

\[ f(t) = 850.00 \quad \text{for } 5 < t \leq 15 \]

### Case 3: \( t > 15 \) (In excess of 15 minutes)
- The professional fee is 850.00 plus an additional 50.00 per minute for each minute over 15 minutes.

\[ f(t) = 850.00 + 50(t - 15) \quad \text{for } t > 15 \]

### Complete Function
Putting it all together, the function \( f(t) \) representing the physician's professional fee based on the time of consultation is:

\[
f(t) =
\begin{cases}
500.00 & \text{if } 0 < t \leq 5, \\
850.00 & \text{if } 5 < t \leq 15, \\
850.00 + 50(t - 15) & \text{if } t > 15.
\end{cases}
\]

This piecewise function represents how the professional fee changes based on the length of the consultation.

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How would the function change if the initial fee for the first 5 minutes were different?
2. What if the additional charge per minute in excess of 15 minutes was 100 instead of 50?
3. How can you graph this piecewise function to visualize the relationship between time and fee?
4. How does the function behave as \( t \) increases indefinitely?
5. What is the significance of the slope in the third piece of the function?

**Tip:** Piecewise functions are useful when different rules apply to different intervals of the domain, as in this scenario with time-based charges.

Understand how to represent a physician's professional fee function based on different time intervals of consultation. This problem involves piecewise functions and linear functions, analyzing changes in fee rates over specified time intervals.

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