What would the maximum area of a rectangular field that is enclosed by 2400 m of fencing?

I have to determine the maximum area of a rectangular field that can be eclosed by 2400 m of fencing. I have no idea what I'm supposed to do. Someone please help me! It's due on Monday.|||Let sides of rectangular are a and b, then


2(a+b) =2400


a+b =1200


a =1200-b


area=A = a*b = (1200-b)*b =1200b -b^2


A' =1200-2b


for maximum area A' =0


1200-2b =0


2b =1200


b =600 m


a = 1200-600=600 m


maximum area = 600*600 =360000 square metres.|||I'll give you 2 ways.





1) The maximum area of a rectangle occurs when that rectangle is a square.





In your problem, that means each side is 600m, which means the area is 360000m虏.





2) Form an equation.


P = 2(l + w) = 2400


----%26gt; l + w = 1200


----%26gt; l = 1200 - w





A = w(1200 - w)


A = -w虏 + 1200w





axis of symmetry --%26gt; x = -b/2a = 600


Use this to find y-coordinate of vertex.





(600, 360000)





The maximum occurs when(x-coordinate) x = 600m and is(y-coordinate) 360000m虏.|||The field is a rectangle, which means it has length and width. Say W is the width and L is the length.





The perimeter of the rectangle is 2W + 2L. The area is WL.





2W + 2L = 2400m


2L = 2400m - 2W


L = 1200m - W





So now you can replace L in the area with that expression.





A = WL = W(1200m - W)


A = -W^2 + (1200m)W





This equation is a parabola, pointed downward; think of A as y and W as x. To find the center point of a parabola in the form of y = ax^2 + bx + c, you take x = -b/(2a). Here a=-1 and b=1200m, so W = -1200m / -2 = 600m. That means L = 1200m - W = 600m, so the rectangle is a square. The area of this square is (600m)(600m) = 360,000 m^2.|||Suppose one side is x; then the other is 1200-x; So the area is f(x)=x*(1200-x)=1200x-x^2. This is a parabola.


It's maximum is there where f'(x)=0; f'(x)=1200-2x. f'(x)=0 when x=600. So it is a square. And the surface is 360.000 m^2.|||a square has the largest area for a given perimeter. So if you have 2400m/ 4sides = 600m sides. Then, multiply the side by the side to get area- 600*600=360,000m^2.|||perimeter = 2400 m


perimeter = 2x+(2400 m-2x)


Area=x(1200 m-x)


f(x)=x(1200 m -x)


max{f(x) = x (1200 m -x)} = 360000 m虏 at x = 600 m|||A rectangle has maximum area for a given perimeter when all 4 sides are congruent. Or say when it's a square


so the max area for 2400 m fencing is


(2400/4)^2=360000 sq m