Answer:
Dimensions would guarantee that the garden has the greatest possible area = 220 feet x 110 feet
Step-by-step explanation:
Fencing available = 440 feet
Here 3 sided need to be considered
That is
Perimeter = 2x Length + width = 2l +w
440 = 2l +w
w = 440 - 2l
We have area = length x width
A = lw
A = l x (440 - 2l) = 440l - 2l²
For area to be maximum we have
[tex]\frac{dA}{dl}=0\\\\\frac{d}{dl}\left ( 440l - 2l^2\right )=0\\\\440-4l=0\\\\l=110feet[/tex]
Substituting in w = 440 - 2l
w = 440 - 2 x 110 = 220 feet
Dimensions would guarantee that the garden has the greatest possible area = 220 feet x 110 feet
The perimeter of a garden is the addition of all side lengths of the garden, while the area is the product of the dimension.
Given that:
[tex]P = 440[/tex] --- the perimeter
Let
[tex]L \to Length \\ W \to Width[/tex]
Because one part is covered by the river, the perimeter is calculated as:
[tex]P = 2L + W[/tex]
So, we have:
[tex]2L + W = 440[/tex]
Make W the subject
[tex]W = 440 - 2L[/tex]
The area (A) of the garden is:
[tex]A = L \times W[/tex]
Substitute [tex]W = 440 - 2L[/tex]
[tex]A = L \times (440 - 2L)[/tex]
[tex]A = 440L - 2L^2[/tex]
Differentiate and set A' to 0
[tex]A' = 440 - 4L[/tex]
Set to 0
[tex]440 - 4L =0[/tex]
Collect like terms
[tex]4L = 440[/tex]
Divide by 4
[tex]L = 110[/tex]
Recall that:
[tex]W = 440 - 2L[/tex]
[tex]W = 440 - 2 \times 110[/tex]
[tex]W = 220[/tex]
Hence, the dimension that would guarantee the greatest possible area is: 110ft by 220ft
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