What are the dimensions that requires least amount of fence if a side of a rectangular field has a stonewall.The area of a rectangular field is 30...

If we need to determine a maximum or minimum amount of
something, we have to create a function that depends on the
amount.

After creating the function, we'll differentiate
it. Then, we'll calculate the roots of the first derivative. These roots, if they exist,
represent the extremes of a function.

In our case, we need
to determine the minimum amount to be fenced.

We'll choose
as amount one dimension of the rectangle. We'll choose the length and we'll note it as
x.

We'll find the other dimension of the rectangle, namely
the width, using the formula of area.

A =
l*w

We'll substitute area by
30.

30 = x*w

We'll divide by
x:

w = 30/x

 Since the
perimeter of the rectangle is calculated with respect to x, we'll create the function
that depends on x, to determine the minimum amount to be
fenced.

The formula of the perimeter of a rectangle
is;

P = 2(l+w)

We'll create
the function:

P(x) = 2(x +
30/x)

Now, we'll
differentiate

P'(X) = (2x +
60/x)'

P'(X) = 2 -
60/x^2

We'll calculate the solution of
P'(X).

P'(X) = 0

2 - 60/x^2 =
0

60/x^2 = 2

2x^2 - 60 =
0

We'll divide by 2:

x^2 - 30
= 0

x^2 = 30

x1 =
+sqrt30

x2 = -sqrt30 is rejected because a length cannot be
negative.

P(sqrt30) = 2sqrt30+
60/sqrt30

P(sqrt30) = (60 +
60)/sqrt30

P(sqrt30) =
120/sqrt30

P(sqrt30) =
120sqrt30/30

P(sqrt30) =
4sqrt30

P(sqrt30) = 4sqrt30 cm is the least
amount to be fenced.