A rectangular field is to have an area of and is to be surrounded by a fence. The cost of the fence is dollars per meter of length. What is the minimum cost this can be done for?.

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

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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?