A forest fire is buring 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 $4000 per square mile. Each person hired is paid $12per hour, and it costs $30 to trasnport and supple each person with the appropriate equipment. Develop a model for determining how many people should be sent to contain the fire and determining the best location for the firebreak.
To determine how many people should be sent to contain the fire and the best location for the firebreak, we need to consider the rate at which the fire is spreading, the time it takes for one person to clear the firebreak, the cost of hiring and transporting the workers, and the value of the lost timber.
First, let's calculate the rate of the fire's spread:
The narrow valley is 3 miles wide, which is equivalent to 3 * 5280 = 15,840 feet wide.
The fire is spreading at a rate of 40 feet per minute.
Next, let's calculate the time it takes for one person to clear one foot of the firebreak:
It takes 30 seconds for one person to clear one foot of the firebreak.
Next, let's determine the area of the firebreak needed:
Since the valley is 15,840 feet wide, the area of the firebreak needed would be the width (15,840 feet) times the length (the time it takes for the fire to reach the other side of the valley in minutes).
Now, let's calculate the time it takes for the fire to reach the other side of the valley in minutes:
Since the valley is 15,840 feet wide and the fire spreads at a rate of 40 feet per minute, the time it takes for the fire to reach the other side is 15,840 / 40 = 396 minutes.
Let's calculate the area of the firebreak needed:
The area = width * length = 15,840 feet * 396 minutes = 6,278,240 square feet.
Now, let's convert the area to square miles:
Since 1 square mile is equal to 27,878,400 square feet, the area of the firebreak needed in square miles is 6,278,240 / 27,878,400 = 0.225 square miles.
Next, let's calculate the value of the lost timber:
The value of the lost timber is $4000 per square mile, and the firebreak needed is 0.225 square miles. Therefore, the value of the lost timber is 0.225 * $4000 = $900.
Now, let's calculate the cost of hiring and transporting the workers:
The workers are paid $12 per hour, and it takes 30 seconds to clear one foot of the firebreak. The time it takes to clear the entire firebreak can be calculated as (length of the firebreak needed in feet) * (time to clear one foot in hours).
The time to clear the entire firebreak = 15,840 feet * 30 seconds * (1 hour / 3600 seconds) = 0.132 hours.
The cost of one worker is ($12 per hour * 0.132 hours) + $30 for transportation and supplies = $31.584.
To determine the number of workers needed, we divide the area of the firebreak by the time it takes for one worker to clear the entire firebreak:
Number of workers needed = area of the firebreak needed / (length of the firebreak needed in feet * time to clear one foot in hours).
Number of workers needed = (15,840 feet * 396 minutes) / (15,840 feet * 30 seconds * (1 hour / 3600 seconds)) = 22.4 workers.
Since we cannot have a fraction of a worker, we round up to the nearest whole number.
Therefore, it is recommended to send 23 workers to contain the fire.
Now, as for determining the best location for the firebreak, it is ideal to cut the firebreak across the narrowest point of the valley. However, without explicit information about the topography and any potential obstacles in the valley, it is difficult to determine the exact best location. A site visit or further analysis of the area would be necessary to determine the optimal location for the firebreak.