Question

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 5 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

Length of enclosure =18.89 foot

Width of enclosure=11.12 foot

Explanation:

We are given that a fence is to be built to enclose a rectangular area 210 square feet.

Fence along three sides is to be made of material that costs 5 dollars per foot

and the material for the fourth side costs 12 dollars per foot.

We have to find the dimension of the enclosure that is most economical

Let x be the length and y be the width of enclosure

We know that area of rectangle=
`x*y`

`xy=210`

`y=(210)/(x)`

Cost of four sides =
`2(5x)+5(y)+12(y)`

Total cost=
`10x+17y`

C=
`10x+17\cdot(210)/(x)`

C=
`10x+(3570)/(x)`

Differentiate w.r.t x

`(dC)/(dt)=10-(3570)/(x^2)`

Substitute
`(dC)/(dx)=0`

`10-(3570)/(x^2)=0`

`(3570)/(x^2)=10`

`x^2=357`

`x=√(357)`

x=18.89

Differentiate w.r.t x

`(d^2C)/(dx^2)=(7140)/(x^3)`

Substitute x=18.89

Then we get

`(d^2C)/(dx^2)=(7140)/((18.89)^3) >0`

Hence, the cost is minimum.

Length of enclosure =18.89 foot

Width of enclosure=
`(210)/(18.89)=11.12 foot`

Hence, the dimension of the enclosure that is most economical to construct

Length=18.89 foot and width=11.12 foot

Answer 2

Answer with explanation:

Let the length of rectangular fence = x meters

Let the width of the rectangular fence = y meters

Thus

`Area=x* y=210ft^(2)...........(i)`

Now Let the total cost to construct the fence be 'C' can be obtained if we construct the shorter side with material with cost $12 per foot

`\therefore C=5* (2y+x)+12* x\\\\C=10y+17x...............(ii)`

Using value of x from the equation i into equation ii we get

`C=10y+17* (210)/(y)`

hence to minimize the cost we differentiate both sides with respect to 'y' and equate the result to zero thus we get

`C=10y+17* (210)/(y)\\\\(dC)/(dy)=(d)/(dy)(10y+(3570)/(y))\\\\0=10-(3570)/(y^(2))\\\\\therefore y=\sqrt{(3570)/(10)}=18.894feet`

Thus
`x=(210)/(18.894)=11.11feet`

Thus value of length = 11.114 feet and value of width = 18.894 feet.