dimensions of rectangular plots

Question

Given that a farmer has 2000 linear feet of fencing,divides a rectangular plot into three nonzero area rectangular subplots,fencing the external perimeter,and separates the three subplots with a fence,answer the following in maximizing the rectangular plot's area

1) dimensions of the rectangular plot

2)dimensions of the three rectangular subplots 

3) maximum area of the rectangular plot

Answer 1

Hi,

 

I assume you are taking Calculus. Ok so let's get started:

 

 

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Let the length of the big rectangle be equal to X, and the width of the big rectangle equal to y

 

then :   X+X+y+y+y+y = 2000

 

so  2x+4y=2000   which means  x+2y =1000

 

Now,

 

Area of the big rectangle is:   A = X.Y

 

but we know from the previous equation (which we found using the perimeter) that x= 1000-2y

 

so substitute that into the area equation so we deal with only one variable:

 

A(y) = (1000-2y)*(y) = 1000y - (2*(y^2))

 

Now we just simply take the derivative, set it equal to 0, to find the max value.

 

A'(y) = 1000 -4y

 

A'(y) =0  =====> 4y=1000  ===>  y=250

 

so plug this value back for y and find x:

 

x=1000- 2y = 1000-500 = 500

 

so when x=500 and y = 250, the area will be maximized. and the area of the large rectangle would be: 500*250 = 125000