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 determine how many people should be sent to contain the fire and the best location for the firebreak, we can develop a mathematical model and implement it on a spreadsheet using Solver.

Step 1: Define the variables and parameters

Let's define the following variables and parameters:
- P: Number of people to be sent
- W: Width of the valley (in feet) = 3 miles * 5280 feet/mile = 15840 feet
- S: Speed of the fire (in feet per minute) = 40 feet/minute
- R: Rate of clearing firebreak (feet per second)

Step 2: Determine the time it takes to clear the firebreak

To find the time it takes to clear the firebreak, we can use the formula:
Time = Width of the valley / Rate of clearing firebreak

Since it takes 30 seconds for one person to clear one foot of the firebreak, the rate of clearing firebreak for one person will be:
R = 1 foot / 30 seconds = 1/30 feet per second

Therefore, the time it takes to clear the firebreak will be:
Time = W / (R * P)

Step 3: Determine the time it takes for the fire to reach the other side of the valley

The time it takes for the fire to reach the other side of the valley can be calculated using the formula:
Time = Width of the valley / Speed of the fire

Therefore, the time it takes for the fire to reach the other side of the valley is:
Time = W / S

Step 4: Develop the objective function

The objective function will be to minimize the cost, which consists of the cost of personnel and the value of lost timber. The cost of personnel can be calculated as:
Cost of personnel = P * (Hourly wage + Cost of transport and supplies) * Time

The value of lost timber can be calculated as:
Value of lost timber = (W / 5280) * (W / 5280) * Cost per square mile

The total cost will be:
Total cost = Cost of personnel + Value of lost timber

Step 5: Implement the model on a spreadsheet

Create a spreadsheet with the following columns:
- Number of people (P)
- Time to clear the firebreak
- Time for the fire to reach the other side
- Cost of personnel
- Value of lost timber
- Total cost

Enter the formulas to calculate the time to clear the firebreak, the time for the fire to reach the other side, and the cost of personnel based on the formulas derived in steps 2 and 4.

Enter the value of the cost per square mile for lost timber.

Use the Solver add-in in Excel to set the objective function to minimize the total cost by changing the number of people (P).

Step 6: Find the optimal solution using Solver

Use Solver to find the minimum total cost by changing the value of P. Solver will determine the number of people that should be sent to contain the fire and the corresponding total cost.

Note: Make sure to set appropriate constraints if necessary, such as the number of people being a positive integer.

By running the Solver, you will obtain the optimal solution, i.e., the number of people to be sent to contain the fire and the corresponding total cost.