The waiting time ,t in mixture, for the `n^(th)` person is a queue, is given by the relation : `t = 12.5n - 15`. If each person taken 10 minutes of be serviced what , is the minutes , between when the services for a person is completed and the service for the next person begins ?

Options

A. 1.5
B. 2.5
C. 3
D. 4

Answer

To solve the problem, we need to find the time difference between when the service for the nth person is completed and when the service for the (n+1)th person begins.

### Step-by-step Solution:

1. **Identify the waiting time for the nth person**:
The waiting time \( t_n \) for the nth person is given by the formula:
\[ t_n = 12.5n - 15 \]

2. **Calculate the waiting time for the (n+1)th person**:
The waiting time \( t_{n+1} \) for the (n+1)th person can be calculated as:
\[ t_{n+1} = 12.5(n + 1) - 15 \]
Simplifying this gives:
\[ t_{n+1} = 12.5n + 12.5 - 15 = 12.5n - 2.5 \]

3. **Find the difference in waiting times**:
Now, we find the difference between the waiting times of the (n+1)th person and the nth person:
\[ t_{n+1} - t_n = (12.5n - 2.5) - (12.5n - 15) \]
Simplifying this, we get:
\[ t_{n+1} - t_n = -2.5 + 15 = 12.5 \]

4. **Determine the time taken for servicing**:
Each person takes 10 minutes to be serviced. Therefore, during the 12.5 minutes between the end of the nth person's service and the start of the (n+1)th person's service, 10 minutes are used for servicing the nth person.

5. **Calculate the remaining time**:
The remaining time after servicing the nth person is:
\[ 12.5 - 10 = 2.5 \text{ minutes} \]

Thus, the time between when the service for the nth person is completed and when the service for the (n+1)th person begins is **2.5 minutes**.

### Final Answer:
The answer is **2.5 minutes**.