A farmer wants to construct a rectangular pen next to a barn

A farmer wants to construct a rectangular pen next to a barn 60 feet long, using all of the barn as part of one side of the pen, the horizontal side. He then wants to divide the pen into three sections by adding two vertical rows of fencing to the pen. If the cost of the horizontal fencing is $4 per foot and the vertical fencing costs $2.50 per foot, find the minimum cost the fencing if the area of the big pen is 16,000 square feet.

Solution

Though, the given data is a bit inadequate, still the question may be answered by making a few assumptions regarding the unclear data in the question first, as follows:

(1) There is a single pen which is to be constructed, so the question of big pen does not arise. Therefore, the pen that has to be constructed will be of 16000 square feet area.

(2) Further, it appears that horizontal fencing would not be required, in general, unless there is a possibility of having some underground constructions, which is not the case here as appears from the given data. Therefore in order to have minimum cost, horizontal fencing should not be used.

The question now becomes straight forward i.e., to find the cost of covering a continuous line enclosing a rectangular region of given area and of given length of one side. It may clearly be done by first finding the perimeter of the rectangle first and then multiplying it by the cost per unit length of the fencing.

Now, if a rectangular pen is to be constructed using 60 feet long barn as part of one side of the pen, and having 16000 square feet area, its second side would be (16000/60) i.e., (800/3) feet.

The perieter of rectangle= {60+(800/3)} feet

Also, as the cost of vertical fencing is $2.5 per foot, the required minimum cost of fencing (without using any horizontal fencing) would be = (2600/3)*2.5= $ 2166.67

3) As given, the cost of vertical and horizontal fencing