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Homework Help: Minimising the perimeter of a tunnel with a fixed area

  1. Oct 19, 2009 #1
    1. The problem statement, all variables and given/known data
    A tunnel cross-section is to have the shape of a rectangle surmounted by a semicircular roof. The total cross-sectional area must be A, but the perimeter minimised to save building costs. Find its dimensions

    2. Relevant equations

    3. The attempt at a solution
    I have that the perimeter would equal pi*r + 4r + 2h (where r is the radius of semi circle, h is height of rectangle) and that A=0.5pi*r^2 + 2rh, but am unsure as to how to get started. Any help would be appreciated.
  2. jcsd
  3. Oct 19, 2009 #2


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    Staff Emeritus
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    Gold Member

    The fact that area must be equal to 'A' has given you a second equation. From it, you can determine the required relationship between r and h...meaning that the problem is reduced down to one variable (either r or h). Finding the value of that one variable that minimizes the perimeter is then just a differential calculus problem.
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