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Optimization problem

  1. Jan 15, 2014 #1
    1. The problem statement, all variables and given/known data

    A farmer has 2400 feet of fencing and want to fence of a
    rectangular field that borders a straight river. He needs no fence along the river.
    What are the dimensions of the field that has the largest area ?

    2. Relevant equations

    We wish to maximize the area A of the rectangle. Let x and y be the width and length of the rectangle (in feet). Then we express A in terms of x and y as A = xy.
    We want to express A as a function of just one variable, so we eliminate y by expressing it in terms of x. To do this we use the given information that the total length of the fencing is 2400 ft. Therefore 2x + y = 2400
    Hence y = 2400 − 2x and the area is A= x (2400 – 2x) = 2400 x − 2x2
    Note that x ≥ 0 and x ≤ 1200 (otherwise A < 0). So the function that we
    wish to maximize is
    A (x) = 2400 x − 2x2, 0 ≤ x ≤ 1200.
    3. The attempt at a solution

    A′(x) = 2400 − 4x, so to find the critical numbers we solve the equation 2400 − 4x = 0 which gives x = 600. The maximum of A must occur either at this critical number or at an end point of the interval. Since A(0) = 0, A(600) = 7,20,000 and A(1200) = 0, thus the maximum value is A (600) = 720,000. When x = 600, y = 2400 − 1200 = 1200.


    but my teacher insists me to solve the problem using second derivative test

    so,A''(x) = − 4

    after this what should we do?
    since the second derivative of x is negative so it is a local maximum
    similarly the second derivative of y is also negative
    so how to find the x and y values?
     
  2. jcsd
  3. Jan 15, 2014 #2

    DrClaude

    User Avatar

    Staff: Mentor

    Note that ##A'(x)## has a single root, therefore it is either the maximum you are looking for, or there is no solution (except for ##x \rightarrow \pm\infty##). You therefore only need to show that ##x=600## is a maximum. Your teacher wants you to do that using ##A''(x)##. You obtain the final answer the same way you already have.
     
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