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A fence is to be built to enclose a rectangular area of 210 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 15 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.
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Step 1Let the length of the rectangular fence be x and breadth be y.So, the area of the fence will be A=xy.The given area is 210 sq. ft.So, xy=210                                                                     1It is given that cost of material for three sides is $3 per foot and for one side is $15 per foot.So, it would be more economical if the fence on shorter side costs $15 per foot.Let the total cost for construction be C(x).Evaluate the cost function as follows.C=3×2x+y+15×xC=6x+3y+15xC=21x+3y                                                                2Step 2Solve (2) using (1) as follows.C=21x+3yC=21210y+3yC=4410y+3yEvaluate the critical point as follows.C'=0ddy4410y+3y=0-4410y2+3=04410y2=3y2=1470y=38.34057y≈38.34 ftThus, y=38.34 ft. Step 3Evaluate x as follows.xy=210x=210yx=21038.34x=5.477308x≈5.48 ftTherefore, the dimensions which are most economical are length is 5.48 ft and breadth is 38.34 ft. ...