A forest fire is burning down a narrow valley 3 miles wide at a speed of 40 feet per minute. The fire can be contained by cutting a firebreak through the forest across the valley. It takes 30 seconds for one person to clear one foot of the firebreak. The value of lost timber is $4,000 per square mile. Each person hired is paid $12 per hour, and it costs $30 to transport and supply each person with the appropriate equipment.
Develop a model for determining how many people should be sent to contain the fire and for determining the best location for the firebreak.
Implement your model on a spreadsheet and find the optimal solution using solver
To develop a model for determining how many people should be sent to contain the fire and the best location for the firebreak, we need to consider several factors: the speed of the fire, the time it takes to clear the firebreak, the cost of hiring and supplying workers, and the value of lost timber.
We will first calculate the time it takes for the fire to reach the other side of the valley. Since the valley is 3 miles wide and the fire is burning at a speed of 40 feet per minute, the time required is (3 miles * 5280 feet/mile) / 40 feet/minute = 396 minutes.
Next, we need to determine how many workers should be sent to contain the fire. Since it takes 30 seconds for one person to clear one foot of the firebreak, the clearing rate is 1 foot per 0.5 minutes or 2 feet per minute. We can divide the length of the firebreak (3 miles * 5280 feet/mile) by the clearing rate of 2 feet/minute to find the number of workers needed.
Number of workers = (3 miles * 5280 feet/mile) / (2 feet/minute) = 7,920 workers.
However, this number is not practical as it is highly unlikely that thousands of workers can be deployed simultaneously. Therefore, to find the optimal number of workers, we can use a spreadsheet and implement the model using Solver, which is a tool available in most spreadsheet software.
To set up the model on a spreadsheet, create the following columns:
1. Number of workers
2. Total time to clear the firebreak (in minutes)
3. Cost of workers' wages
4. Cost of transportation and supplies
5. Total cost
6. Value of lost timber
Fill in the values for the number of workers starting from a reasonable number (e.g., 100) and increment by a certain value (e.g., 10) until we reach a time that exceeds the time it takes for the fire to reach the other side of the valley.
Calculations for each row:
- Total time to clear the firebreak = (Length of firebreak) / (Clearing rate of 2 feet/minute)
- Cost of workers' wages = (Number of workers) * (Time to clear the firebreak in hours) * ($12 per hour)
- Cost of transportation and supplies = (Number of workers) * ($30 per worker)
- Total cost = Cost of workers' wages + Cost of transportation and supplies
- Value of lost timber = (Area of the valley) * ($4,000 per square mile)
Finally, use Solver to find the optimal number of workers by minimizing the total cost while considering the constraint that the total time to clear the firebreak should not exceed the time it takes for the fire to reach the other side of the valley.
Once Solver finds the optimal number of workers, you can analyze the results to determine if it is practical to deploy that many workers. If it is not, you may need to adjust the model and assumptions accordingly.
Note: The best location for the firebreak was not mentioned in the given information. To determine the best location, additional factors such as topography, vegetation density, wind direction, and other terrain characteristics need to be considered.