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Verdich [7]
3 years ago
13

A fence is to be built to enclose a rectangular area of 240 square feet. The fence along three sides is to be made of material t

hat costs 6 dollars per foot, and the material for the fourth side costs 12 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.
Mathematics
1 answer:
aniked [119]3 years ago
8 0

Answer:

18.97 \text{ and }12.65\text{ft}

Step-by-step explanation:

GIVEN: A fence is to be built to enclose a rectangular area of 240 square feet. The fence along three sides is to be made of material that costs 6 dollars per foot, and the material for the fourth side costs 12 dollars per foot.

TO FIND: Find the dimensions of the enclosure that is most economical to construct.

SOLUTION:

Area of rectangular fence =240\text{ sq. ft}

let the length of fence =x

let the width of fence =y

let x be the smaller side

Area of rectangular fence enclosure =xy

xy=240

y=\frac{240}{x}

cost of fence along three sides =6 \text{ dollars per feet}

cost of fence along fourth side =12 \text{ dollars per feet}

length of fence =2x+2y

cost of fence building =6(2x+y)+12y

                                     =12x+18y

putting value of y

                                     =12x+18\times\frac{240}{x}

                                     =12x+4320x^{-1}

to find minimum value differentiating the equation

                                   12-\frac{4320}{x^{2} }=0

                                   x^{2} =\frac{4320}{12}

                                   x^{2} =360

                                   x=18.97\text{ft}

                                   y=12.65\text{ft}

Hence the dimensions of the enclosure that is most economical to construct are 18.97\text{ft} and 12.65\text{ft}

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