You are giving 24 linear feet of fencing all 3 feet tall you wish to make the largest rectangular enclosure possible to maximize space what should the dimensions of the enclosure be

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Answer:

The largest area enclosed is  A = xy = 6 feet [tex]\times[/tex] 6 feet = 36 [tex]feet^{2}[/tex]

Step-by-step explanation:

i) The perimeter of the area is 2[tex]\times[/tex](x + y) =24  ∴ x + y = 12    ∴ y = 12 - x

ii) The area of rectangle enclosed A  = xy    ⇒ A  = x ( 12 - x) = 12x - [tex]x^{2}[/tex]

iii) differentiating both sides of the equation in ii) we get

[tex]\dfrac{dA}{dx} = 12 - 2x = 0[/tex]    ⇒  x = 6 feet

iv) Differentiating both sides of equation in iii) we get    [tex]\frac{d^{2}A}{dx^{2} }[/tex] =  -2

   Therefore the area enclosed is maximum as the double derivative is negative

v) therefore for largest area enclosed   x  = 6 feet and  y = 12 - 6 = 6 feet

vi) therefore the largest area enclosed is

 A = xy = 6 feet [tex]\times[/tex] 6 feet = 36 [tex]feet^{2}[/tex]

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