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A farmer wishes to enclose a rectangular field that is already bounded on one side by a river (this side needs no fence). What are the dimensions of the field of maximum area that can be enclosed with 300 meters of fence)? 

 Nov 7, 2016

Best Answer 

 #1
avatar+33661 
+5

Let length = L, width = W and area = A

 

L + 2W = 300

 

A = L*W

 

A = (300 - 2W)*W  or  A = 300W - 2W^2

 

Using calculus:

dA/dW = 300 - 4W  This is zero when W = 75, hence L = 300 - 2*75  or L = 150

 

Without calculus:

Completing the square:

A = -2(W^2 - 150W) → -2( (W - 75)^2 -75^2 )   This is a clearly a minimum when W = 75, hence L = 150.

.

 Nov 7, 2016
 #1
avatar+33661 
+5
Best Answer

Let length = L, width = W and area = A

 

L + 2W = 300

 

A = L*W

 

A = (300 - 2W)*W  or  A = 300W - 2W^2

 

Using calculus:

dA/dW = 300 - 4W  This is zero when W = 75, hence L = 300 - 2*75  or L = 150

 

Without calculus:

Completing the square:

A = -2(W^2 - 150W) → -2( (W - 75)^2 -75^2 )   This is a clearly a minimum when W = 75, hence L = 150.

.

Alan Nov 7, 2016
 #2
avatar+33661 
+5

I should have said "maximum" not "minimum" in the sentence below Completing the square!

Alan  Nov 7, 2016

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