a farmer wants to construct a fence around a rectangular field. Sides with neigh

Question

a farmer wants to construct a fence around a rectangular field. Sides with neighbours need reinforced fencing that cost 6$ per meter. The other sides use regular fencing that cost 3$ per meter. Assuming the farmer has neighbours on the west and south sides, answer the following quesions with (showing your work).

a) given a budget of 975$, find the dimensions that give the maximum area inside the fencing:

-Length of the east and west fences

-Length of the north and the south fences

b) Find the dimensions that will cost the least to build a fence with an area of 1410 m^2

-Length of the east and west fences

-Length of the north and the south fences

Explanation / Answer

a) let length of east ,west fences be x , north south fences be y

budget =975

(6x+3y+ 3x+6y)=975

9x+9y =975

x+y =975/9

y=(975/9)-x

area A=x*y

A=x*((975/9)-x)

A=(975/9)x-x2

area is maximum when dA/dx =0 ,d2A/dx2<0

dA/dx =(975/9)-2x=0

2x=975/9

x=975/18

x=54.17

d2A/dx2=-2<0

y=(975/9)-(975/18)

y=54.17

length of east west fences =54.17m,

length of north south fences=54.17m

======================================================

b)area xy =1410

y =1410/x

cost C=(6x+3y+ 3x+6y)

cost C=9(x+y)

C=9(x+(1410/x))

for minimum cost dC/dx =0 ,d2C/dx<0

dC/dx =9(1-(1410/x2))=0

1-(1410/x2)=0

x2=1410

x=37.55m

d2C/dx =9(0+(2*1410/x3))<0

9(0+(2*1410/37.553))<0

y =1410/x

y =37.55m

cost to build is minimum when

length of east west fences =37.55m,

length of north south fences=37.55m

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