Question

Need to make a rectangular pen for pigs that will enclose a total area of 169 square feet. What is the least amount of fencing that will be needed?

Answer 1

The least amount of fencing needed for the rectangular pen is 72.19 feet.

Explanation:

The area and perimeter equations of the pen are, respectively:

`p = 2\cdot (x + y)` (1)

`A = x\cdot y` (2)

Where:

`p` - Perimeter, in feet.

`A` - Area, in square feet.

`x` - Width, in feet.

`y` - Length, in feet.

Let suppose that total area is known and perimeter must be minimum, then we have a system of two equations with two variables, which is solvable:

From (2):

`y = (A)/(x)`

(2) in (1):

`p = 2\cdot \left(x + (A)/(x)\right)`

And the first and second derivatives of the expression are, respectively:

`p' = 2\cdot \left(1 -(A)/(x^(2)) \right)` (3)

`p'' = (4\cdot A)/(x^(3))` (4)

Then, we perform the First and Second Derivative Test to the function:

First Derivative Test

`2\cdot \left(x - (A)/(x^(2)) \right) = 0`

`2\cdot \left((x^(3)-A)/(x^(2)) \right) = 0`

`x^(3) - A = 0`

Given that dimensions of the rectangular pen must positive nonzero variables:

`x^(3) = A`

`x = \sqrt[3]{A}`

Second Derivative Test

`p'' = 4`

In a nutshell, the critical value for the width of the pen leads to a minimum perimeter.

If we know that
`A = 169\,ft^(2)`, then the value of the perimeter of the rectangular pen is:

`x = \sqrt[3]{169\,ft^(2)}`

`x \approx 5.529\,ft`

By (2):

`y = (A)/(x)`

`y = (169\,ft^(2))/(5.529\,ft)`

`y = 30.566\,ft`

Lastly, by (1):

`p = 2\cdot (5.529\,ft + 30.566\,ft)`

`p = 72.19\,ft`

The least amount of fencing needed for the rectangular pen is 72.19 feet.