Respuesta :
Answer:
Therefore the dimension of the rectangular piece of land is 375 m by 250 m.
Step-by-step explanation:
Given that,
Two opposite sides of rectangular land will be fenced using standard fencing that costs $6 per m while the other two side will require heavy duty fencing that costs $9 per m.
Let the length and width of the rectangular x and y respectively.
Then total cost for fencing = [tex]\$(2x\times 6+2y\times 9)[/tex]
=$(12x+18y)
Then,
12x+18y= 9,000
⇒ 2x+3y=1,500 [ cancel out common factor 6]
⇒3y= 1500 - 2x
[tex]\Rightarrow y=\frac{1500-2x}{3}[/tex] ....(1)
The area of the rectangular land is = Length × width
= xy
Let,
A=xy
Putting the value of y
[tex]A=x.\frac{1500-2x}{3}[/tex]
[tex]\Rightarrow A=\frac13(1500x-2x^2)[/tex]
Differentiating with respect to x
[tex]A'=\frac13(1500-4x)[/tex]
Again differentiating with respect to x
[tex]A''=-\frac43[/tex]
Now we have to determine the critical value of A.
For critical value set A'=0
[tex]\frac13(1500-4x)=0[/tex]
⇒1500-4x=0
[tex]\Rightarrow x=\frac{1500}{4}[/tex]
[tex]\Rightarrow x=375[/tex] m
Since, [tex]A''|_{x=375}=-\frac43<0[/tex]. at x=375 m the area of the rectangular piece of land is maximum.
From equation (1) we get when x=375
[tex]y=\frac{1500-2\times 375}{3}[/tex]
⇒y=250 m
Therefore the dimension of the rectangular piece of land is 375 m by 250 m.
The dimension of the rectangular lot of greatest area is [tex]250m \ by \ 375m[/tex].
The perimeter of rectangle:
The formula for the perimeter of a rectangle is,
[tex]P=2(l+b)[/tex]
Let [tex]L[/tex] be the length & B the breadth in meters.
Perimeter[tex]=2L+2B[/tex]
[tex](6\times2L+9\times2B)=9000[/tex]
[tex]L=750-1.5B[/tex]
[tex]\Rightarrow Area \ A=LB=(750-1.5B)\\=750B-1.5B^2[/tex]
[tex]{A}'=750-3B \ and \ {A}''=-3[/tex] (i.e. negative, so setting [tex]{A}'[/tex] to zero will give a maxima)
[tex]0=750-3B \\ B=250m \\ L=750-1.5\times 250 \\ =375m[/tex]
Learn more about the Perimeter of the rectangle:
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