farmer has 96 meters of fencing- he wants to enclose a rectangular field and build a fence across the middle, parallel to the shorter ends. ?

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farmer has 96 meters of fencing- he wants to enclose a rectangular field and build a fence across the middle, parallel to the shorter ends. ?

(a) express the total area of the field as a function of width w, where w is the length of the shorter ends;
(b) algebraically determine the value of w for which the area is a maximum.

Posted 268 day ago

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Answers (3)

let w be the shrt side of the rectangle, and y be the long side. so 3w + 2y = 96m.
or y = (96-3w)/2
area of rectangle = A = wy= w(96-3w)/2 =
48w - (3w^2)/2
in order for A to be maximum, its derivative must be zero. so
A' = 0 = 48 -3w or 3w = 48 or w = 16.
so 3w = 2y = 96 or 48 = 2y, or y=24. the rectangle is 16 by 24, its area is 192 m^2, a maximum.

Posted 231 day ago

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Amar Soni

Perimeter of rectangle+length of the fence=2L+3w=96 therefore L=(96-3w)/2.=48-3w/2..........(i)
Area of rectangle= L.w=w{(96-3w)/2= 48w-3(w^2)/2}
For Maximum Area differentiate
dA/dw= 48w-6w/2=48w-3w
For maximum or minimum Area dA/dw=0
Therefore 48-3w=0 or d=16
L= 48-(3*16)/2=48-24=24......................(ii)
Maximum Area= L*W=24*12=384 sq. units

Posted 250 day ago

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tutorsTeach

function of x (the field area) is: 48w - 3/2 w^2
To get to that you will realize: 3 w's are needed to build the 2 shorter ends plus the one in the middle because the farmer wants to split the field.
The longer ends are each (96-3w)/2. We divide by two because it is the length of each side of the rectangle.
Then (96-3w)/2 times w (w being the shorter side of the rectangle):
w[(96-3w)/2], and that is the formula of this particular rectangle area.

The area would be the largest if it is square because the multipliation of equal numbers make a larger sum than the multiplication of one smaller than the other (e.g., 3x3=9 but 2x4=8 and 5x1=5). What we do, we divide 96 meters by 5, then multiply:
96/5 = 19.2
19.2x19.2 = 368.64 square meters
That would make the largest field (rectangle/square) area.

Posted 267 day ago

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