Melissa is building a rectangular area for her cow, Daisy. She plans to build the area along a straight stretch of river (no fence is needed along the river).
a. Melissa would like to give daisy 800m^2 of grazing room. What is the least amount of fencing she will need?
b. She realizes that she only has 60 m of fencing. What is the largest grazing area she will be able to give daisy?
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Answered by
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least amount of fencing needed = 20 +20+40 = 80 m
largest grazing area by 60m fencing = 250m²
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Answered by
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Let us assume that
- Length of Rectangular area = 'x' m
and
- Breadth of Rectangular area = 'y' m
So,
- According to statement,
Now,
- Length of fencing, L = x + 2y
So,
Differentiating both sides w. r. t. y, we get
For maxima or minima,
Now,
Again differentiating (2) w. r. t. y, we get
Now,
Substituting the value of y in equation (1), we get
Hence,
The fence is shortest if the side parallel to the river has length 40 m and the other 2 sides each have length 20 m.
Let us assume that
- Length of Rectangular area = 'x' m
and
- Breadth of Rectangular area = 'y' m
So,
- According to statement,
Length of fencing = 60 m
Now,
Differentiating both sides w. r. t. y, we get
For, maxima or minima,
Now, again differentiating both sides w. r. t. y, we get
Hence, Area is maximum.
So,
Substituting the value of y from equation (3) in (1), we get
Hence,
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