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.

To develop a model for determining how many people should be sent to contain the fire and for determining the best location for the firebreak, we need to consider a few factors:

1. Time taken to cut the firebreak: Since it takes 30 seconds for one person to clear one foot of the firebreak, we can calculate the time taken to clear the entire width of the valley. We know that the valley is 3 miles wide, which is equivalent to 3 miles * 5,280 feet/mile = 15,840 feet. Considering the speed of the fire at 40 feet per minute, we can calculate the time taken to reach the other side of the valley as 15,840 feet / 40 feet per minute = 396 minutes.

2. Timber loss: The value of lost timber is given as $4,000 per square mile. Since we are dealing with a narrow valley, we need to calculate the area that needs to be protected or saved from the fire. The area of the valley can be calculated by multiplying the width (3 miles) by the depth (unknown). Let's use the variable "d" to represent the depth of the valley. So, the area to be saved is 3 miles * d. Multiplying this by the value of lost timber ($4,000 per square mile), the cost of the lost timber will be 4,000 * (3 miles * d).

3. Cost of the firebreak crew: The cost of the firebreak crew includes the wages and the transportation and equipment cost per person. The wages for each person are $12 per hour. Since we know that it takes 396 minutes (6.6 hours) to clear the firebreak, we can multiply this by the hourly wage to get the total wages for the crew. The transportation and equipment cost is given as $30 per person. So, the cost of the crew will be 6.6 hours * $12 per hour + $30 per person.

Now, to determine the number of people needed, we need to compare the time taken to cut the firebreak with the time available to contain the fire. If the time taken to cut the firebreak is less than the time available to contain the fire, only a single crew will be required. If it is greater, we will calculate the number of crews needed based on the time available.

The best location for the firebreak can be determined by analyzing the geography of the valley, the wind direction, and any natural barriers in the path of the fire. Selecting a location that minimizes the area of forest that needs to be saved can also be a factor.

By considering these factors and performing the calculations, we can develop a model to determine the number of people needed to contain the fire and the best location for the firebreak.