When a forrest fire occurs, how many fire fighter should be sent to fight the fire so that the total cost is kept minimum?
Suppose the fire starts at ; at , the fire fighter arrives; the fire is extinguished later at .
Let and denotes the monetary damage per square footage burnt, the hourly wage per fire fighter, the one time cost per fire fighter and the number of fire fighters sent respectively. The total cost consists damage caused by the fire, the wage paid to the fire fighters and the one time cost of the fire fighters:
(total square footage burnt)
Notice , the duration of fire fighting.
Assume the fire ignites at a single point and quickly spreads in all directions with flame velocity , the growing square footage of engulfed circular area is a function of time. Namely,
Its burning rate
However, after the arrival of fire fighters, is reduced by , an amount that is directly proportional to the number of fire fighters on the scene. The reduction of reflects the fire fighting efforts exerted by the crew. As a result, for declines along the line described by
where . Or equivalently,
Moreover, the fire is extinguished at suggests that
Combine (1) and (3),
It is further assumed that
is continuous at
We illustrate in Fig. 1.
The fact that is continuous at means
The area of triangle in Fig. 1 represents , the total square footage damaged by the fire. i.e.,
Consequently, the total cost
To minimize the total cost, we seek a value where function attains its minimum value .
From expressing as
It follows that
Hence, to minimize is to find the that minimizes
Since , a constant, by the following theorem:
For positive quantities and positive rational quantities , if is a constant, then attains its minimum if
(see “Solving Kepler’s Wine Barrel Problem without Calculus“), we solve equation
From Fig. 2, we see that , contradicts (2).
However, when is a positive quantity. Therefore, it is
that minimizes the total cost.