During the campus Spring Fling, the bumper car amusement attraction has a problem of cars becoming disabled and in need of repair. Repair personnel can be hired at the rate of $20 per hour, but they only work as one team. Thus, if one person is hired, he or she works alone; two or three people work together on the same repair.

 
     One repairer can fix cars in an average time of 30 minutes. Two repairers take 20 minutes, and three take 15 minutes. While these cars are down, lost income is $40 per hour. Cars tend to break down at the rate of 1.60 per hour.


What is the total hourly cost with one repair person? (Round your intermediate calculations and final answer to 2 decimal places.)

To calculate the total hourly cost with one repair person, we need to account for both the cost of hiring the repair person and the lost income due to cars being disabled. 

### Step 1: Calculate the repair time with one repair person
One repairer takes an average of 30 minutes to fix a car. Converting this to hours:

\[
\text{Time per repair} = \frac{30}{60} = 0.5 \text{ hours per car}.
\]

### Step 2: Calculate the repair rate (repairs per hour) with one repair person
The repair rate is the reciprocal of the time it takes to repair one car. Therefore, the repair rate is:

\[
\text{Repair rate} = \frac{1}{0.5} = 2 \text{ cars per hour}.
\]

### Step 3: Calculate the utilization of the repair person
The utilization is the ratio of the breakdown rate to the repair rate. The breakdown rate is given as 1.60 cars per hour:

\[
\text{Utilization} = \frac{1.60}{2} = 0.80.
\]

This means the repair person is busy 80% of the time.

### Step 4: Calculate the average number of cars in the system
The system has a breakdown rate of 1.60 cars per hour and a repair rate of 2 cars per hour. The average number of cars in the system (using a basic queuing theory formula for an M/M/1 system) is:

\[
L_q = \frac{\text{Utilization}}{1 - \text{Utilization}} = \frac{0.80}{1 - 0.80} = 4.
\]

### Step 5: Calculate the average waiting time per car
The waiting time is the average number of cars in the system divided by the breakdown rate:

\[
W_q = \frac{L_q}{1.60} = \frac{4}{1.60} = 2.5 \text{ hours}.
\]

### Step 6: Calculate the lost income
For every hour a car is down, the lost income is $40 per hour. The total lost income is the product of the number of cars waiting and the lost income per car:

\[
\text{Lost income} = 4 \times 40 = 160 \text{ dollars per hour}.
\]

### Step 7: Calculate the cost of hiring the repair person
The cost of hiring one repair person is $20 per hour.

### Step 8: Calculate the total hourly cost
The total hourly cost includes both the lost income and the cost of the repair person:

\[
\text{Total hourly cost} = 160 + 20 = 180 \text{ dollars per hour}.
\]

Thus, the total hourly cost with one repair person is **$180.00**.

Would you like more details on any step or have any other questions?

---

Here are 5 related questions:
1. How does the cost change if two repairers are hired?
2. What is the total cost of lost income if the cars break down at a different rate?
3. How does the repair rate affect the total hourly cost?
4. What is the total waiting time with two or three repairers?
5. How can queue management reduce the waiting time and cost?

**Tip:** Consider using queuing theory models (e.g., M/M/1 or M/M/c) to analyze similar situations for optimal resource allocation.

During the campus Spring Fling, the bumper car amusement attraction has a problem of cars becoming disabled and in need of repair. Repair personnel can be hired at the rate of $20 per hour, but they only work as one team. Thus, if one person is hired, he or she works alone; two or three people work together on the same repair. One repairer can fix cars in an average time of 30 minutes. Two repairers take 20 minutes, and three take 15 minutes. While these cars are down, lost income is $40 per hour. Cars tend to break down at the rate of 1.60 per hour. What is the total hourly cost with one repair person?

Calculate the total hourly cost of hiring one repair person for the bumper car attraction at a Spring Fling event. This analysis uses queuing theory and cost analysis, covering repair rates, breakdown rates, and utilization to determine the lost income and repair cost. Suitable for undergraduate students studying operations research or applied mathematics.

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