Question

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

Answer 1

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 $12.65\text{ft}$