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Math Help - Midterm...Calculus optimization problem

  1. #1
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    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!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Dubulus View Post
    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 n of them, and all the stands are the same width.

    Then there are n+1 stands and the total width of the forrest is:

     <br />
6=(n+1)w+n \times 0.1 {\rm{km}}<br />

    where w is the width of a stand. So:

    w=\frac{6-n \times 0.1}{n+1}

    The the lost area is:

     <br />
A(n)=w \times 4 + 4 \times n \times 0.1=4 \times \frac{6-n \times 0.1}{n+1}+4 \times n \times 0.1<br />

    Which is what you have to minimise.

    CB
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