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Sagot :
The minimum cost for the area can be fenced is $1920
It is given that the area of the rectangular field is 900 m²
Let the length of one side of the field be x m
Therefore, the length of the other side will be
900/x
The perimeter of the rectangular field
= 2(length + breadth)
= 2(x + 900/x)
= 2x + 1800/x m
Therefore, the cost of fencing will be
$(32x + 28800/x )
Let f(x) = 32x + 28800/x
We need to minimize f(x)
Differentiating with respect to x and equating it to 0 gives us
32 - 28800/x² = 0
or, x² = 28800/32
or, x² = 900
or, x = 30
This is because the length cannot be negative hence -30 cannot be a value
Now
f''(x) = 57600/x³
f''(30) = 2/1333
Since f"(x) is positive, x = 30 is the minimum value of x for which the cost is minimum.
Hence, the minimum cost will be
f(30)
= 960 + 960
= $1920
To know more about minimization visit
https://brainly.com/question/21286880
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Complete Question
A rectangular field is to have an area of 900 m² and is to be surrounded by a fence. The cost of the fence is 16 dollars per meter of length. What is the minimum cost this can be done for?
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