a rectengular field 21 cm^2 in area needs to be fenced. find the dimensions that requires least amount of fence if one side is protected by barn.

Whenever we need to determine a maximum or minimum amount
of something, we need to create a function that depends on the amount. After creating
the function, we'll calculate it's first derivative. 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
21.

21 = x*w

We'll divide by
x:

w = 21/x

Now, we'll create
the function that depends on x, to determine the minimum amount to be fenced. This
function is the perimeter of the rectangle.

The formula of
the perimeter of a rectangle is;

P =
2(l+w)

We'll create the
function:

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

Noe, we'll calculate the first
derivative.

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

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

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

P'(X) = 0

2 - 42/x^2 =
0

42/x^2 = 2

2x^2 - 42 =
0

We'll divide by 2:

x^2 - 21
= 0

x^2 = 21

x1 =
+sqrt21

x2 = -sqrt21

P(sqrt21)
= 2sqrt21+ 42/sqrt21

P(sqrt21) = (42sqrt21 +
42sqrt21)/21

P(sqrt21) = 4sqrt21 cm is the
least amount to be fenced.