157k views
1 vote
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.

2 Answers

6 votes

Answer:

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

User Jason Hoffmann
by
9.2k points
5 votes

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.

User Mikko Ohtamaa
by
7.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories