Dimensions.What are the dimensions that requires least amount of fence if a side of a rectangular field is a wall. The area of a rectangular field is 30 cm^2.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.