A farmer wants to fence a rectangular area of 128 square feet next to a river. Find the length and width of the rectangle which uses the least amount of fencing if no fencing is needed along the river. Assume the length of the fence runs parallel to the river.

For least amount of fencing, length and width of rectangular area should be 16 feet and 8 feet respectively.

Let us consider length and width are l and w respectively.

 Area of rectangular region =

Since, no fencing needed along the river.

So, perimeter of rectangular region, P =

Substituting the value of w in above expression.

We get,                        

For least amount of fencing, perimeter should be minimum

Differentiate perimeter expression with respect to length l.

So,  

Thus, For least amount of fencing, length and width of rectangular area should be 16 feet and 8 feet respectively.

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Length=16 ft

Width=8 ft

Step-by-step explanation:

Let width be w and length will be 128/w since area is product of length and width

Perimeter will be w+w+128/w. Note that along the river, fencing isn't needed that is why perimeter equals 2w+128/w

Getting the first differential of perimeter with respect to w then

At critical point

Therefore, since length is 128/w, length is 128/8=16 ft

Therefore, maximum length and width are 16 ft and 8ft respectively

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