Question

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

Answer 1

Answer:

The largest area enclosed is  A = xy = 6 feet $\times$ 6 feet = 36 $feet^{2}$

Step-by-step explanation:

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

ii) The area of rectangle enclosed A  = xy    ⇒ A  = x ( 12 - x) = 12x - $x^{2}$

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

$\dfrac{dA}{dx} = 12 - 2x = 0$    ⇒  x = 6 feet

iv) Differentiating both sides of equation in iii) we get    $\frac{d^{2}A}{dx^{2} }$ =  -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 $\times$ 6 feet = 36 $feet^{2}$