Thread: Midterm...Calculus optimization problem

A company owns a plot of land with a forest on it. They want to install firebreaks, which are sections of the forest that are cut out. this way if a fire starts, it won't destroy the whole forest. anyways. the land is 4 miles by 6 miles. the firebreaks will be evenly spaced, and are .1miles long. find the best number n which will minimize (using calculus) the area lost by a fire. that is, minimize the sum of the area of forest in one stand of trees and the total area of all firebreaks.
then minimize the revenue lost if a fire occurs. it costs $15,000 per square mile for creating the firebreaks. areas of the forest unaffected by the fire are worth $40,000. below is an example of the forest where n=3. any help is greatly appreciated, especially the work for a solution!

Originally Posted by Dubulus

A company owns a plot of land with a forest on it. They want to install firebreaks, which are sections of the forest that are cut out. this way if a fire starts, it won't destroy the whole forest. anyways. the land is 4 miles by 6 miles. the firebreaks will be evenly spaced, and are .1miles long. find the best number n which will minimize (using calculus) the area lost by a fire. that is, minimize the sum of the area of forest in one stand of trees and the total area of all firebreaks.
then minimize the revenue lost if a fire occurs. it costs $15,000 per square mile for creating the firebreaks. areas of the forest unaffected by the fire are worth $40,000. below is an example of the forest where n=3. any help is greatly appreciated, especially the work for a solution!

Lets suppose the firebreaks run accros the forrest as in the diagram, and suppose there are of them, and all the stands are the same width.

Then there are stands and the total width of the forrest is:



where is the width of a stand. So:



The the lost area is:



Which is what you have to minimise.

CB