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Math OPTIMIZATION

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A home gardener plans to enclose two rectangular gardens with fencing. The dimensions of the garden: x by 12-x, y by 12-x-y
a. Find the values of x and y that maximize the total area enclosed.
b. What is the maximum total area enclosed?
c. How many meters of fencing are needed?

  • Math OPTIMIZATION - ,

    The total area A=A(x,y).
    A=x(12-x)+y(12-x-y)=12x-x^2+12y-xy-y^2
    Partial derivatives
    A'x=12-2x-y
    A'y=12-x-2y

    Solve the equations A'x=A'y=0 => a. x=y=4
    b. Amax=32+16=48
    c. 24+16=40

  • Math OPTIMIZATION - ,

    hey thanks! but where did you get 16 in b?

  • Math OPTIMIZATION - ,

    y(12-x-y)=4(12-4-4)=16

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